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ON THE INVERSE OF PARABOLIC SYSTEMS OF PARTIAL

DIFFERENTIAL EQUATIONS OF GENERAL FORM IN AN

INFINITE SPACE–TIME CYLINDER

THOMAS KRAINER AND BERT–WOLFGANG SCHULZE

Abstract. We consider general parabolic systems of equations on the infi-nite time interval in case of the underlying spatial configuration is a closedmanifold. The solvability of equations is studied both with respect to timeand spatial variables in exponentially weighted anisotropic Sobolev spaces,and existence and maximal regularity statements for parabolic equations areproved. Moreover, we analyze the long-time behaviour of solutions in termsof complete asymptotic expansions.

These results are deduced from a pseudodifferential calculus that we con-struct explicitly. This algebra of operators is specifically designed to containboth the classical systems of parabolic equations of general form and theirinverses, parabolicity being reflected purely on symbolic level. To this end,we assign t = ∞ the meaning of an anisotropic conical point, and prove thatthis interpretation is consistent with the natural setting in the analysis ofparabolic PDE. Hence, major parts of this work consist of the constructionof an appropriate anisotropic cone calculus of so-called Volterra operators.

In particular, which is the most important aspect, we obtain the completecharacterization of the microlocal and the global kernel structure of the in-verse of parabolic systems in an infinite space–time cylinder. Moreover, weobtain perturbation results for parabolic equations from the investigation ofthe ideal structure of the calculus.

Contents

Introduction 3

Chapter 1. Preliminary material 131.1. Basic notation and general conventions 13

Functional analysis and basic function spaces 14

Date: January 21, 2002.1991 Mathematics Subject Classification. Primary: 35K40; Secondary: 35A05, 35A17, 35B40,

35B65, 35Sxx, 58J40.Key words and phrases. Parabolic partial differential equations, time–dependent problems,

global solvability and regularity, asymptotics of solutions, structure of solution operators, pertur-bation theory, pseudodifferential operators, Volterra calculus, anisotropic totally characteristicoperators.

The first author was supported by the Deutsche Forschungsgemeinschaft.

1

2 T. KRAINER AND B.–W. SCHULZE

Preliminaries on function spaces and the Mellin transform 16Global analysis 19

1.2. Finitely meromorphic Fredholm families in Ψ-algebras 201.3. Volterra integral operators 27

Some notes on abstract kernels 30

Chapter 2. Abstract Volterra pseudodifferential calculus 332.1. Anisotropic parameter-dependent symbols 33

Asymptotic expansion 34Classical symbols 35

2.2. Anisotropic parameter-dependent operators 36Elements of the calculus 37Ellipticity and parametrices 39Sobolev spaces and continuity 42Coordinate invariance 42

2.3. Parameter-dependent Volterra symbols 43Kernel cut-off and asymptotic expansion of Volterra symbols 44The translation operator in Volterra symbols 46

2.4. Parameter-dependent Volterra operators 47Elements of the calculus 48Continuity and coordinate invariance 49Parabolicity for Volterra pseudodifferential operators 50

2.5. Volterra Mellin calculus 53Continuity in Mellin Sobolev spaces 57

2.6. Analytic Volterra Mellin calculus 58Elements of the calculus 61The Mellin kernel cut-off operator and asymptotic expansion 62Degenerate symbols and Mellin quantization 64

2.7. Volterra Fourier operators with global weight conditions 66

Chapter 3. Parameter-dependent Volterra calculus on a closed

manifold 703.1. Anisotropic parameter-dependent operators 70

Ellipticity and parametrices 763.2. Parameter-dependent Volterra operators 79

Kernel cut-off behaviour and asymptotic expansion 81The translation operator in Volterra pseudodifferential operators 84Parabolicity for Volterra operators on manifolds 85

Chapter 4. Weighted Sobolev spaces 864.1. Anisotropic Sobolev spaces on the infinite cylinder 864.2. Anisotropic Mellin Sobolev spaces 90

Mellin Sobolev spaces with asymptotics 924.3. Cone Sobolev spaces 96

ON THE INVERSE OF PARABOLIC PDE IN INFINITE SPACE–TIME 3

Chapter 5. Calculi built upon parameter-dependent operators 995.1. Anisotropic meromorphic Mellin symbols 995.2. Meromorphic Volterra Mellin symbols 106

Mellin quantization 1095.3. Elements of the Mellin calculus 110

Ellipticity and Parabolicity 1145.4. Elements of the Fourier calculus with global weights 119

Ellipticity and Parabolicity 122

Chapter 6. Volterra cone calculus 1276.1. Green operators 1276.2. The algebra of conormal operators 131

Operators that generate asymptotics 131Calculus of conormal symbols 133The operator calculus 136Smoothing Mellin and Green operators 148

6.3. The algebra of Volterra cone operators 150The symbolic structure 158Compositions and adjoints 161

6.4. Ellipticity and Parabolicity 166Parabolic reductions of orders 176

Chapter 7. Remarks on the classical theory of parabolic PDE 177

References 182

Introduction

Parabolic partial differential equations arise in the modelling of time-dependentphenomena, e. g., in the description of diffusion processes, such as heat diffusion,as well as in probability, where we often find probability densities of stochasticprocesses as solutions of associated parabolic equations or systems, as it is thecase for (certain) Markov chains. Moreover, there are deep connections between theanalysis of the heat equation associated with geometric operators and (spectral)geometry.

In general, external influences, such as exterior sources, and interactions withthe geometry of the underlying spatial configuration, lead to non-autonomousequations or systems, i. e., the coefficients may depend on time, and we haveto solve them with inhomogeneous data.

To give an example, consider the heat diffusion flow in a body, representedby a closed manifold X . Then, as the diffusion u interacts with the geometry ofX , perturbed by an exterior source f , we find it as a solution of the heat equation(∂t −∆gX (t)

)u = f , where gX(t) is a family of Riemannian metrics, depending on

time.


The analysis of parabolic partial differential equations is concerned, in par-ticular, with the following questions:

• Existence and uniqueness of solutions in appropriate function spaces.• Local properties of solutions, such as regularity and local bounds for the

derivatives, on a finite time interval.• Global properties, e. g., global bounds and/or integrability conditions, as

well as stabilization of the solution and its derivatives, especially asymp-totics, on the infinite time interval.

In this work, we consider general parabolic systems of equations on the infinite timeinterval in case of the underlying spatial configuration is a closed manifold. Thesolvability of equations is studied both with respect to time and spatial variablesin exponentially weighted anisotropic Sobolev-Slobodeckij spaces (which will becalled just Sobolev spaces in the sequel). In particular, this leads to fine analysisof regularity, which results into existence and maximal regularity statements forparabolic equations. Moreover, we analyze the long-time behaviour of solutions interms of complete asymptotic expansions.

These results are deduced from the concept of regularity of a pseudodifferen-tial calculus that we construct explicitly. This algebra of operators is specificallydesigned to contain both the classical systems of parabolic equations of generalform and their inverses, parabolicity being reflected purely on symbolic level. Tothis end, we assign t = ∞ the meaning of an anisotropic conical point, and provethat this interpretation is consistent with the natural setting in the analysis of par-abolic PDE (see [34]). Hence, major parts of this work consist of the constructionof an appropriate anisotropic cone calculus of so-called Volterra operators.

In particular, which is the most important aspect of this work, we obtainthe complete characterization of the microlocal and the global kernel structure ofthe inverse of parabolic systems on the infinite time interval. Moreover, we obtainperturbation results for parabolic equations from the investigation of the idealstructure of the calculus.

Let us enter this subject in more detail with some historical and methodicalremarks first.

Elements of the classical theory. In parabolic partial differential equationsthere is a canonical splitting of variables into the (preferred) time and the spatialvariables, and the analysis requires the anisotropic treatment of these. The mostdirect approach is based upon the anisotropic treatment of space and time on sideof the covariables. Parabolicity in this framework is regarded as some “strong”anisotropic ellipticity with the time covariable polynomially rescaled in order tocompensate the different orders between spatial and time derivatives. This pointof view goes back essentially to the classical works of Petrovskij [48].

A deep connection between parabolicity and anisotropic parameter-depen-dent ellipticity is given via regarding the time covariable as a parameter for theoperators acting in space. This was observed and systematically exploited by Agra-novich and Vishik [2] in their work about boundary value problems for parabolic


partial differential equations of general form. The solvability of equations is es-tablished both with respect to time and spatial variables in anisotropic Sobolevspaces. This framework is particularly well-suited for the analysis of regularity ofdistribution solutions.

As general references in this context we want to mention the works ofEjdel’man [14], Ejdel’man and Zhitarashu [15], Friedman [16], Ladyzhenskaya,Solonnikov, and Uraltseva [38], Lions and Magenes [40], and Solonnikov [66], [67],[68].

The concept of anisotropic parameter-dependent ellipticity also plays an im-portant role in spectral theory and the analysis of resolvents (see, e. g., Shubin [65]).This can be regarded as a link between the microlocal approach, i. e., anisotropictreatment of space and time on side of the covariables as just discussed, and thesemigroup-theoretical approach to parabolic equations, which is, roughly speak-ing, based upon anisotropic treatment on side of the variables (see, e. g., Amann[3], Lunardi [41], Pazy [47], and Tanabe [69]). The solution there is given in termsof an evolution operator, and can be seen in the context of singular (Volterra)integral operators, where the fundamental solution plays the role of an operator-valued convolution kernel. Hence, starting from a partial differential equation, thesolution operator is rather implicit due to the emphasis of the kernel level, and themicrolocal character is not reflected. We shall not pursue this discussion further.

The global long-time behaviour of solutions is an important feature in thestudy of equations posed on the infinite time interval. The analysis is essentiallydevoted to establish bounds at infinity, and it is most natural to ask for solutionssatisfying exponential estimates (see, e. g., Agranovich and Vishik [2]); of course,such can be expected only under suitable assumptions on the coefficients of theequation as well as on the inhomogeneous data. More refined control of the globalbehaviour of solutions is reflected by asymptotic stabilization, or even asymptoticexpansions. Asymptotic analysis of partial differential equations is for itself a fieldof independent interest in mathematics with a long tradition. Concerning long-time behaviour and exponentially stable solutions we just want to mention theworks of Agmon and Nirenberg [1], Maz’ya and Plamenevskij [42], and Pazy [46],[47]. However, a sufficiently complete analysis including perturbation theory of thelong-time asymptotical behaviour for solutions to parabolic equations seems notto be available in the literature yet, even under rather strong assumptions on thecoefficients of the equations and the inhomogeneous data.

Pseudodifferential analysis of parabolic equations. The basic idea in pseu-dodifferential analysis in general is to embed differential operators, which are “typ-ical” for a certain problem setting, into an algebra of operators with symbolicstructure, and to study, e. g., the solvability and regularity of equations therein.The symbolic structure plays the dominating role in all investigations, and condi-tions and manipulations on the microlocal side are reflected on the operator level– the quantized objects – usually up to a small ideal of residual elements thatqualitatively can be neglected in the considerations in question.


This concept is particularly well-established in the theory of elliptic equations,where ellipticity is determined by the invertibility of the symbolic components, andthe existence of parametrices within the calculus is proved. In particular, Fredholmsolvability for elliptic equations is achieved in natural scales of Sobolev spaces, withthe parametrix being a Fredholm inverse. The analysis of the operators in thealgebra, applied to the parametrix and the remainders, provides detailed ellipticregularity results, including the asymptotic behaviour of solutions near the singularsets in the theory of degenerate elliptic equations. Moreover, this naturally impliesperturbation results for elliptic equations.

See, e. g., Boutet de Monvel [5], Grubb [25], Rempel and Schulze [53] forboundary value problems with the transmission property, and Schulze [59], [60],[61] for the more general case of pseudodifferential theory of degenerate ellipticoperators, where the degeneracy reflects the presence of geometric singularities onthe underlying manifold in a typical way.

In contrast to elliptic theory, we classically expect unique solvability for par-abolic equations. However, it is still mostly desirable to have the achievements ofelliptic theory at hand also in the framework of parabolicity, i. e., to take advantagein the study of equations from a specifically designed calculus of pseudodifferentialoperators. Hence, the program to be carried out is the following:

• Completion of the most natural systems of non-autonomous parabolicequations of general form to an enveloping algebra of pseudodifferentialoperators.

• Characterization of parabolicity purely on symbolic level by means of theinvertibility of the symbolic components.

• Proof of the equivalence of symbolic and operational invertibility, i. e.,parabolic operators are invertible, and the inverses belong to the calculus.

• Representation of the algebra as bounded operators acting in the naturalscale of anisotropic Sobolev spaces (with an exponential weight at infinity).

• Extension of the concept of regularity for the calculus in the sense, thatthe analysis of both smoothness (via the smoothness-parameters of theSobolev spaces) and asymptotics (via subspaces that carry the asymp-totic information) of solutions is a consequence of the general mappingproperties of the operators in these spaces.

As a consequence, the microlocal character of the solution operator and its globalkernel structure are clarified, and an extensive study of regularity and global be-haviour of solutions, as well as perturbation theory, is available purely in algebraicterms on side of the algebras of symbols and operators, as well as their idealstructure.

A first step towards this program was done in the works of Piriou [49], [50];however, his approach was not really widely applied. Let us shortly summarize theimportant contents:

He introduced the class of anisotropic scalar pseudodifferential operators withthe Volterra property in order to investigate parabolic pseudodifferential equations


on a finite time interval, where the underlying spatial configuration is a closedmanifold. The significant feature of these operators is that they are built uponanisotropic symbols, the anisotropy referring to time and space, that extend holo-morphically in the time covariable to the lower complex half-plane, including thesymbol estimates. It is proved that this class remains preserved under composi-tions. Parabolicity is defined by the invertibility of the anisotropic homogeneousprincipal symbol, extended to the half-plane, and a parametrix construction iscarried out within the algebra of operators with the Volterra property. Due to thePaley-Wiener theorem, the Schwartz kernel of a Volterra pseudodifferential opera-tor is supported below the diagonal with respect to the time variable; in fact, thisis the justification for this notion. As a consequence, the operators are “one-sidedlocal” with respect to time, i. e., the support of a distribution is preserved by theaction of the operator from the positive side. Hence, a Neumann series argument isapplicable to the kernels of the remainders of the parametrix construction, whichinsures the invertibility of parabolic operators within the calculus, as they areconsidered on the finite time interval only.

Stimulated by Piriou’s results, Rempel and Schulze [53] initiated similar in-vestigations for parabolic boundary value problems, and subsequently first stepswere done by Buchholz [6], and Buchholz and Schulze [7], [8], to approach the caseof the underlying spatial manifold having geometric singularities. However, thesestudies were restricted to problems on the finite time interval, while in the presentwork we fully carry out the above program in case of operators on the infinitetime interval, and thus the analysis of the relevant effects near t = ∞ is included(see also [34]). As turns out, the non-compactness with respect to time is respon-sible for the presence of an additional operator-valued symbol in the regulation ofparabolicity, and the before-mentioned concepts for the calculus on a finite timeinterval have to be extended considerably.

A basic observation to achieve the desired results is, that it is possible tointerpret t = ∞ as a conical point of the infinite space-time configuration, andthis interpretation is consistent with the natural setting in parabolic problems.Let us illustrate this a bit more for differential operators:

Consider an anisotropic differential operator on R×X

A =

µ/ℓ∑

j=0

aj(t)∂jt ,

aj ∈ C∞(R,Diffµ−ℓj(X ;E,F ))

(1)

of general form, where µ ∈ ℓN0 is the anisotropic order of A, and Diffµ−ℓj(X ;E,F )denotes the space of differential operators of order µ−ℓj acting in (smooth) sectionsof the vector bundles E and F on X . The anisotropy ℓ ∈ N refers to the differenttreatment of space and time for the operator A; for the heat operator, e. g., wehave ℓ = 2 and µ = 2.


Writing A in the coordinates r = e−t leads to

A =

µ/ℓ∑

j=0

aj(r)(−r∂r)j ,

aj(r) = aj(− log(r)),

(2)

and the effects near t = ∞ are now located at r = 0. We assume that the coefficientsaj(r) extend as smooth functions up to the origin r = 0 – in the original coordinates

this corresponds to exponential stabilization as t→ ∞ – hence, the operator A canbe regarded as an anisotropic totally characteristic operator. Notice that operatorsA with coefficients not depending on time for t ≫ 0 sufficiently large belong toour setting (see also Agranovich and Vishik [2]).

In singular analysis, operators of the form (2) are widely investigated in theframework of elliptic theory, for this is precisely the form of the typical differentialoperators near a conical singularity. The natural function spaces are Mellin Sobolevspaces, and the anisotropic variants of these are exactly the exponentially weightedSobolev spaces on the cylinder R×X , written in the new coordinates on R+×X .

Hence, for the study of parabolicity of the operator (1), we consider it fromthe very beginning as given in the form (2). Our construction of the envelopingpseudodifferential algebra then relies on techniques which originate from elementsof the cone calculus introduced by Schulze (in an anisotropic setting, see [13], [59],[60], or [61]), and, which is the crucial step, on establishing the analogue of Piriou’soperators with the Volterra property in this framework.

The Volterra cone calculus is given in Chapter 6: Section 6.3 deals with thealgebraic properties, and in Section 6.4 we discuss parabolicity and the invertibilityof the operators within the algebra. As a by-product, we furthermore obtain acalculus for anisotropic elliptic totally characteristic operators, and a Fredholmtheory for these in anisotropic weighted Sobolev spaces.

The concept of regularity of this calculus covers the control of conormalasymptotics, i. e., asymptotic expansions for functions u of the form

u(r, x) ∼∑

j

mj∑

k=0

cj,k(x) logk(r)r−pj , r → 0, (3)

where the pj ∈ C are complex numbers, only finitely many located in every stripparallel to the imaginary axis over a compact real interval, and the cj,k are smoothsections in a vector bundle over X . Equivalently, in the original space-time coor-dinates, these take the form

u(t, x) ∼∑

j

mj∑

k=0

cj,k(x)tkepjt, t→ ∞, (4)

of exponential long-time asymptotics as desired.


Organization of the text and further comments. In Chapter 1 we give anaccount on the notations, and shortly summarize some preliminary material, e. g.,about the Mellin transform, that we freely use throughout this work.

Due to the role played by meromorphic operator functions in the symboliccalculus of the final algebra as to control the asymptotic behaviour of solutions(see, in particular, Sections 5.1, 5.2, and 6.2), we decided to supply Section 1.2,where we recall the classical theorem on the inversion of finitely meromorphicFredholm families (see Gohberg and Sigal [19], Gramsch [20], and Gramsch andKaballo [22]). This theorem is used in the construction of parametrices of parabolicoperators, more precisely in symbolic inversion.

Moreover, we recall in Section 1.3 in some detail elements of the theoryof Volterra integral operators with operator-valued L2-kernel functions. Providedthat the kernel is continuous and fulfills suitable weighted estimates we give a proofthat the associated operator is quasinilpotent, i. e., its spectrum consists of zeroonly. This observation is crucial for the analysis of remainders of the parametrixconstruction to parabolic pseudodifferential operators, and leads to the invertibilityof these within the calculus.

We conclude the chapter with some notes on abstract kernels. The mappingproperties of an operator within a scale of suitable function spaces is closely relatedto the behaviour of its Schwartz kernel; in applications, the residual elements of apseudodifferential calculus are usually characterized by such mapping properties.In order to be able to apply the inversion result for Volterra integral operatorsto these operators, we have to conclude that the Schwartz kernels satisfy certainweighted estimates. From the abstract point of view, the relationship betweenmapping properties and kernels is given by means of tensor product representa-tions.

Chapter 2 is devoted to recall some basic elements of pseudodifferential calcu-lus with operator-valued symbols. In general, a global calculus of pseudodifferen-tial operators is built upon underlying structures of operator-valued symbols, e. g.,passing to local coordinates in the interior of a manifold gives rise to matrix-valuedsymbols. Moreover, in parameter-dependent calculi we often find residual elementscharacterized as operator-valued symbols within a suitable scale of Sobolev spaces,while the parameter-dependent calculus itself embeds into a space of operator-valued symbols. Consequently, we find operator-valued symbols both as a sub-and superstructure, which enables us to trace back many global constructions tothe abstract calculus of pseudodifferential operators with operator-valued symbols.

Though some technical properties of our Volterra cone calculus could bededuced in a more direct way, we prefer to make as much use of the abstractsetting given in Chapter 2 as possible. On the one hand, this shows that themore complicated constructions later are in fact based upon some few analyticprinciples that we formulate explicitly, and on the other hand, it demonstratesthat our methods should be extendable to apply to more complicated situations,e. g., parabolic equations with geometric singularities on the spatial configuration.


Intuitively, the abstract calculus considered in Sections 2.1 – 2.4 should bethought of as operators acting in spatial direction with the time covariable un-affected as a parameter, while in the remaining sections of Chapter 2 we havethe converse situation, i. e., operators acting in time with the spatial covariablesunaffected. For our purposes, the calculi of Volterra symbols and operators are ofcourse crucial, and many of the basic general constructions in pseudodifferentialcalculus have to be considerably modified to apply to this framework.

A rigorous treatment of the abstract general calculus of Volterra pseudodif-ferential operators with “twisted” operator-valued symbols is given in Krainer [33],[32]; see also Buchholz and Schulze [8]. In [33] and [32] the reader will also findthose details of proofs that were skipped in the present exposition. Material onthe general calculus of pseudodifferential operators with “twisted” operator-valuedsymbols as introduced by Schulze can be found, e. g., in [59], [60], [61].

In Chapter 3 we recall the calculus of anisotropic parameter-dependent pseu-dodifferential operators acting in sections of vector bundles on a closed manifold.The interpretation is that the parameter should be regarded as the time covariable.Moreover, we study the subcalculus of parameter-dependent Volterra operators,where the parameter-space is a complex half-plane, and the operator families de-pend holomorphically on the parameter. We define the corresponding notions ofparameter-dependent ellipticity and parabolicity for such operators, and carry outthe parametrix construction within the (Volterra) calculus.

The definitions and arguments are traced back to the considerations fromSections 2.1 – 2.4; in a local chart, we find operators that are built upon matrix-valued (Volterra) symbols, while the global smoothing remainders are precisely theregularizing operator-valued (Volterra) symbols in the standard Sobolev spaces ofdistributional sections in the bundles. Using elementary norm estimates of theoperators in terms of the parameter, we conclude that the calculus itself embedsinto a suitable space of operator-valued (Volterra) symbols in the Sobolev spaces.This observation, in particular, enables us to add some necessary supplements,such as kernel cut-off, simply via restriction from the abstract setting.

Chapter 4 is devoted to state the basic definitions and properties of theweighted anisotropic Sobolev spaces on the transformed space-time configurationin that form as they are needed in this exposition – the formulation in global termsvia parameter-dependent reductions of orders admits, e. g., to deal in Chapter 5in an efficient way with the continuity of pseudodifferential operators that arebuilt upon symbols which themselves are parameter-dependent operators on themanifold. The elementary analysis of anisotropic Sobolev spaces is widely availablein the literature, cf. Agranovich and Vishik [2], Grubb and Solonnikov [27], Lionsand Magenes [40]). Material about (isotropic) Mellin Sobolev spaces can be foundin the monographs of Schulze, concerning Mellin Sobolev spaces with discreteconormal asymptotics we refer, in particular, to [59]. Finally, we want to pointout that though we employ the notion of “cone Sobolev spaces” due to superficialsimilarities with the corresponding spaces in the analysis on manifolds with conical


singularities, the spaces in our framework are essentially different from these, seeSection 4.3.

In Section 5.1 and 5.2 we introduce certain spaces of meromorphic functionstaking values in the pseudodifferential operators on the manifold, which will laterserve as meromorphic operator-valued (Volterra) Mellin symbols in the final al-gebra near the origin r = 0. We define the notions of ellipticity and parabolicityand prove inversion results under these conditions. In this context, we decided tosupply Section 1.2.

Section 5.3 and 5.4 are concerned with pseudodifferential calculi where thesymbols are built upon parameter-dependent (Volterra) operators on the manifold– these operators now act in function spaces on the full (transformed) space-timeconfiguration. The pseudodifferential properties, such as composition, are conse-quences of the results in the general abstract setting. We define the notion ofparabolicity for Volterra operators and establish the existence of Volterra para-metrices. In fact, the arguments rely on the results for the parameter-dependentcalculus of Chapter 3. In addition, we handle ellipticity for general anisotropicsymbols.

The (holomorphic) Mellin calculus from Section 5.3 is of major importance,for these operators contribute to the final algebra near the origin r = 0, whichcorresponds to t → ∞ in the original coordinates; the calculus of Section 5.4 willbe employed away from r = 0, i. e., near t = −∞. Isotropic meromorphic Mellinsymbols and Mellin pseudodifferential calculus play an important role in the elliptictheory on manifolds with singularities, see [13], [59], [60], [61]; via specializing inSection 5.1 to ℓ = 1, e. g., we find the symbol spaces which are considered inthe cone calculus. For the treatment of parabolicity, however, we have to imposeadditional structures, and a much more refined analysis is required. In isotropicelliptic theory, a global Fourier calculus and an ellipticity criterion for operatorsconsidered in Section 5.4 were obtained by Seiler [64].

In Chapter 6 we establish the Volterra cone calculus. The definition of thecalculus and its symbolic structure, as well as the analysis of the algebraic prop-erties, is given in Section 6.3. Near the origin r = 0 we employ Volterra Mellinoperators with meromorphic symbols, and away from r = 0 we find the operatorsfrom Section 5.4. In addition, there arise Volterra Green operators (Section 6.1)as residual elements. Section 6.4 is devoted to study parabolicity, and to establishthe invertibility of parabolic operators within the calculus. In Section 6.2 we in-troduce an auxiliary algebra which allows us to present the analysis of regularitywith asymptotics (the conormal effects) in a transparent form.

Parabolicity is determined by three symbols: The interior symbol (in the clas-sical case this is just the anisotropic homogeneous principal symbol), the conormalsymbol that controls the effects at r = 0, and the exit symbol which reflects thebehaviour as r → ∞ – all symbols extend in a canonical way holomorphically inthe (transformed) time covariable to a half-plane. Provided that all symbols areinvertible, the invertibility of the operator within the algebra follows; in particular,the operator is invertible in the Sobolev spaces on R+×X .


In parabolic partial differential equations there is usually an initial time t0 ∈R, and the problem is posed on the time interval [t0,∞), which corresponds inour setting to an interval (0, r0] with r0 ∈ R+ – there are no effects as r → ∞.Indeed, if we are just interested in the invertibility of an operator on an intervalof this form, we can drop the parabolicity assumption for the exit symbol and stillfind the inverse operator in the calculus, but now restricted to subspaces of theSobolev spaces which consist of all distributions with support in (0, r0]×X (seeSection 6.4).

We conclude the chapter with a proof of the existence of parabolic reductionsof orders in our calculus; in particular, there are parabolic operators for any givenorder. This result, e. g., simplifies the analysis of parabolic boundary value prob-lems, for they then are reduced to the case of the interior parabolic operator aswell as all boundary conditions having the same unified pseudodifferential order(see [34]).

Finally, in Chapter 7, we give some more remarks about how the classicaltheory of parabolic partial differential equations fits into the framework of ourVolterra cone calculus. To this end, we discuss the classical notion of parabolicity,as well as the results about solvability and regularity, for a generalized heat oper-ator, and draw the connection to the functional analytic structure of our calculusfor this example. In particular, the chapter may be thought of as an additionalguide to the previous chapters along the lines of a particularly simple example.

Concluding remarks and future prospects. The achievements of pseudodif-ferential theory affected the analysis of parabolic equations in various other direc-tions, in particular, what the study of equations of pseudodifferential character isconcerned; see, e. g., Grubb [23], [24], Grubb and Solonnikov [27], Iwasaki [28],and Purmonen [52].

The present work, however, aims at another direction. Though the equationsunder consideration were modelled over a closed manifold, we have proved that,due to the additional non-compactness in time, we ended up with a theory fordegenerate operators. In fact, this instance should be thought of as a generalrule (see also [34]). Thus, in view of the insights from singular analysis that arenowadays available, our results should also be regarded as a step towards singular(or degenerate) parabolic problems, and the more advanced analysis of highersingularities in the future will rely upon them.

Indeed, many interesting and challenging problems from theory and applica-tions belong to the singular problems:

• The non-compatible case of parabolic initial-boundary value problems isof high relevance and natural in models of applications. This is to a largeextent not yet treated in the literature in a sufficiently general form.

• Parabolic mixed boundary problems, e. g., of Dirichlet/Neumann type(i. e., like in Zaremba’s problem), see, for instance, Chan Zui Cho andEskin [9].


• Parabolicity for degenerate cases, e. g., for boundary conditions of type ofthe oblique derivative problem; see Paneah [44], Popivanov and Palagachev[51].

• Parabolicity for geometric singularities of the spatial configuration, e. g.,stratified spaces, where the singularities induce a hierarchy of (operator-valued) symbolic levels, see Schulze [62]. Necessary results in this directioncan be found in Krainer [33], [32].

• Long-time asymptotics for singular spatial configurations, characterizationof adequate asymptotic terms; see, e. g., Krainer and Schulze [35].

Chapter 1. Preliminary material

1.1. Basic notation and general conventions

Sets of real and complex numbers.

• We denote:

C the complex numbers,R the reals,R+,R− the positive (negative) reals,R+,R− the non-negative (non-positive) reals,Z the integers,N the positive integers,N0 the non-negative integers.

• Let CN and RN denote the complex N -space, respectively the EuclideanN -space, in the variables (z1, . . . , zN ) ∈ CN or (x1, . . . , xN ) ∈ RN , respec-tively. In general, we allow N to be zero, and in this case these spacesdegenerate to the set containing a single point only.

• The upper half-plane in C will be denoted as

H := z ∈ C; Im(z) ≥ 0.

Moreover, for β ∈ R let

Γβ := z ∈ C; Re(z) = β,Hβ := z ∈ C; Re(z) ≥ β.

We refer to Γβ also as a weight line. With the splitting z = β+ iτ into realand imaginary part we shall identify Γβ with R via Γβ ∋ z = β + iτ ↔τ ∈ R.

Analogously, we have an identification of Hβ with the right half-planeH0 via Hβ ∋ z = β + ζ ↔ ζ ∈ H0, i. e., Hβ originates from H0 viatranslation, and we shall also employ the identification of Hβ with theupper half-plane H via H ∋ ζ ↔ β − iζ ∈ Hβ.


• The Euclidean norm of x = (x1, . . . , xN ) ∈ RN is denoted as |x| =( N∑j=1

x2j

) 12

. Moreover, let 〈x〉 =(1+ |x|2

) 12 be the standard regularized dis-

tance in RN . The inner product in RN is denoted as 〈x, ξ〉 = xξ =N∑j=1

xjξj .

Multi-index notation. We employ the standard multi-index notation.For multi-indices α = (α1, . . . , αN ), β = (β1, . . . , βN ) ∈ NN0 we denote

(α

β

)=

(α1

β1

). . .

(αNβN

)α! =

N∏

j=1

αj ! |α| =

N∑

j=1

αj .

We write α ≤ β if the inequality holds componentwise. Moreover, (normalized)partial derivatives with respect to the variables x = (x1, . . . , xN ) ∈ RN are writtenas

∂αx =∂|α|

∂α1x1 . . . ∂

αNxN

Dαx = (−i)|α|∂αx .

In case a function f(x, λ) depends on the group of complex variables λ ∈ CM wealso use the notations

∂βλf =∂|β|

∂β1

λ1. . . ∂βMλM

f Dβλf = (−i)|β|∂βλf,

∂βλf =

∂|β|

∂β1

λ1. . . ∂βM

λM

f Dβ

λf = (−i)|β|∂β

λf.

For z = (z1, . . . , zN) ∈ CN and α = (α1, . . . , αN ) ∈ NN0 we write zα = zα11 · . . . ·zαNN .

Functional analysis and basic function spaces. Unless stated explicitly oth-erwise, the spaces in this work are always assumed to be complex.

For topological vector spaces E and F we denote the space of continuous lin-ear operators E −→ F as L(E,F ). Moreover, the topological dual of E is denotedas E′. We write E⊗F for the algebraical tensor product of E and F . The pro-jective and injective topology on E⊗F is indicated by the subscripts E⊗πF andE⊗εF , respectively, while E⊗πF and E⊗εF denote the corresponding comple-tions. We employ the notation 〈·, ·〉E,F , or just 〈·, ·〉, when we deal with a dualityE×F −→ C. The inner product in a Hilbert space E is also denoted as 〈·, ·〉E , orsimply as 〈·, ·〉.

Moreover, we have the following spaces of E-valued functions on M (whereM and E are appropriate):

Lp(M,E) measurable functions u with∫M

‖u(x)‖pE dx <∞(with respect to Lebesgue measure, 1 ≤ p <∞),

C(M,E) continuous functions,A(M,E) analytic functions,Ck(M,E) k-times continuously differentiable functions,


C∞(M,E) smooth functions,C∞

0 (M,E) smooth functions with compact support,C∞b (M,E) smooth functions with bounded derivatives,

S(M,E) rapidly decreasing functions,D′(M,E) = L(C∞

0 (M), E) distributions,E ′(M,E) = L(C∞(M), E) distributions with compact support,S′(M,E) = L(S(M), E) tempered distributions.

If E = C we drop it from the notation.The following spaces of smooth, bounded functions naturally occur in Mellin

pseudodifferential calculus:Let E be a Frechet space. Define

C∞B ((R+)q, E) := u ∈ C∞((R+)q, E);

((−r∂r)ku

)((R+)q) ⊆ E

is bounded for all k ∈ Nq0,

endowed with the Frechet topology of uniform convergence of (−r∂r)ku on (R+)q

for every k ∈ Nq0. Here we use the notation (−r∂r)k := (−r1∂r1)k1 . . . (−rq∂rq)kq

for r = (r1, . . . , rq) ∈ (R+)q and k = (k1, . . . , kq) ∈ Nq0.

Moreover, let C∞B ((R+)q, E) := C∞

B ((R+)q, E) ∩ C∞b ((R+)q, E) be the sub-

space of all functions that extend smoothly to (R+)q.

Hilbert triples and (formal) adjoint operators. A triple E0, E,E1 of Hilbertspaces E0, E,E1 is called a Hilbert triple, if the following conditions are fulfilled:

a) There exists a Hausdorff topological vector space X such that E0, E and E1

are embedded in X .b) E0 ∩ E ∩ E1 is dense in E0, E and E1.c) The inner product on E induces a non-degenerate sesquilinear pairing 〈·, ·〉 :

E0 × E1 → C, that provides antilinear isomorphisms E′0∼= E1 and E′

1∼= E0.

Let E0, E,E1 and E0, E, E1 be Hilbert triples. Then, for each A ∈ L(E0, E0),

there is a unique operator A∗ ∈ L(E1, E1) such that 〈Ae0, e1〉E = 〈e0, A∗e1〉E for

all e0 ∈ E0 and e1 ∈ E1. A∗ is called the (formal) adjoint operator of A.

The mapping A 7→ A∗ provides an antilinear isomorphism L(E0, E0) →L(E1, E1).

Tempered distributions and the Fourier transform. Let E be a Hilbert space.Partial derivatives of a distribution u ∈ S′(Rn, E) are defined as 〈∂αx u, ϕ〉 =(−1)|α|〈u, ∂αxϕ〉, while multiplication with a function ψ of tempered growth isgiven as 〈ψu, ϕ〉 = 〈u, ψϕ〉. A distribution u ∈ S′(Rn, E) is called regular, if u is aBochner measurable function, and there exists N ∈ N0 with

∫Rn

〈x〉−N‖u(x)‖E dx <∞. Note that we identify regular distributions with their densities. In this sensewe in particular have Lp(Rn, E) → S′(Rn, E).


We employ the normalized Fourier transform F : S(Rn) −→ S(Rn), i. e.,

(Fu)(ξ) = (2π)−n2

∫

Rn

e−ixξu(x) dx,

(F−1u)(x) = (2π)−n2

∫

Rn

eixξu(ξ) dξ,

for u ∈ S(Rn). For Frechet spaces E the Fourier transform extends to an iso-morphism S(Rn, E) −→ S(Rn, E) via F = F⊗πidE , noting that S(Rn, E) ∼=S(Rn)⊗πE. If E is a Hilbert space we have F : S′(Rn, E) −→ S′(Rn, E) via〈Fu, ϕ〉 = 〈u,Fϕ〉.

Elementary symbol spaces. For a Frechet space E we denote the space of symbolsof order µ ∈ R with values in E as Sµ(Rn, E), i. e., a function a ∈ C∞(Rn, E)belongs to Sµ(Rn, E) if and only if ∂αξ a(ξ) = O(〈ξ〉µ−|α|) as |ξ| → ∞, for allα ∈ Nn0 .

Similarly, the space Sµ,(Rn×Rn, E) of symbols of order (µ, ) is the space of

all a ∈ C∞(Rn×Rn, E) such that ∂αx ∂βξ a(x, ξ) = O(〈x〉−|α|〈ξ〉µ−|β|) as |x, ξ| → ∞,

for all α, β ∈ Nn0 .These spaces are Frechet spaces in a canonical way. If any one of the orders

equals −∞ in the notations involved, we mean the corresponding intersection overall spaces with finite orders.

Preliminaries on function spaces and the Mellin transform. Let E be aFrechet space.

• For γ ∈ R let

Tγ(R+, E) := u ∈ C∞(R+, E);(r

12−γ〈log(r)〉m(−r∂r)ku

)(R+) ⊆ E

is bounded for all k,m ∈ N0.

This space is endowed with the Frechet topology of uniform convergence

of r12−γ〈log(r)〉m(−r∂r)ku on R+ for every k,m ∈ N0.

Note that for every δ ∈ R the operator of multiplication with the func-tion rδ induces a topological isomorphism rδ : Tγ(R+, E) −→ Tγ+δ(R+, E).

• For γ ∈ R define the operator

Sγ : u(r) 7−→ e(γ−12 )tu(e−t) (1.1.1)

and its inverse

S−1γ : u(t) 7−→ rγ−

12u(−logr). (1.1.2)

The operator (1.1.1) is well-defined as a topological isomorphism

Sγ : D′(R+) → D′(R)


and restricts to topological isomorphisms on various subspaces, e. g.,

Sγ :

C∞

0 (R+) −→ C∞0 (R)

Tγ(R+) −→ S(R).

This shows, in particular, that Tγ(R+) is a nuclear Frechet space withC∞

0 (R+) as a dense subspace, and we have a canonical isomorphismTγ(R+, E) ∼= Tγ(R+)⊗πE.

• The (weighted) Mellin transform (defined on C∞0 (R+))

(Mγu)(z) :=

∫

R+

rzu(r)dr

r

for z ∈ Γ 12−γ

with its inverse

(M−1γ u)(r) :=

1

2πi

∫

Γ 12−γ

r−zu(z) dz

extends via Mγ = Mγ⊗πidE to a topological isomorphism

Mγ : Tγ(R+, E) −→ S(Γ 12−γ

, E).

For u ∈ Tγ(R+, E) we have

Mγ((−r∂r)u)(z) = zMγ(u)(z), Mγ((logr)u)(z) = DτMγ(u)(z),

Mγ+δ(rδu)(z) = Mγ(u)(z + δ).

(1.1.3)

• For u ∈ C∞0 (R+, E) the Mellin transform Mu extends to an entire function

such that Mu|Γ 12−γ

= Mγu, and the mapping

R ∋ γ 7→ Mγu ∈ S(Γ 12−γ

, E)

is a C∞-function taking values in the rapidly decreasing functions.

A Paley–Wiener type theorem. Let E be a Frechet space.

• For t0 ∈ R let

S0((t0,∞), E) := u ∈ S(R, E); supp(u) ⊆ [t0,∞),S0((−∞, t0), E) := u ∈ S(R, E); supp(u) ⊆ (−∞, t0],

as well as

A(H(−), E; t0) := f ∈C∞(H(−), E) ∩ A(

H(−), E);

[H(−) ∋ z 7→ eit0zf(z) ∈ E] ∈ S(H(−), E)with either the upper half-plane H or the lower half-plane H− := z ∈C; Im(z) ≤ 0 in C involved; these spaces are Frechet with the projectivetopology with respect to the mappings

A(H(−), E; t0) ∋ f 7−→ eit0zf(z) ∈ S(H(−), E).


Then the Fourier transform F : S(R, E)∼=−→ S(R, E) restricts to topolog-

ical isomorphisms

F :

S0((−∞, t0), E)

∼=−→ A(H, E; t0)

S0((t0,∞), E)∼=−→ A(H−, E; t0)

for every t0 ∈ R.• For r0 ∈ R+ and γ ∈ R let

Tγ,0((0, r0), E) := u ∈ Tγ(R+, E); supp(u) ⊆ (0, r0],and let

A(H 12−γ

, E; r0) := f ∈C∞(H 12−γ

, E) ∩ A(

H 12−γ

, E);

[H 12−γ

∋ z 7→ r−z0 f(z) ∈ E] ∈ S(H 12−γ

, E).The latter is a Frechet space with the projective topology with respect tothe mapping

A(H 12−γ

, E; r0) ∋ f 7−→ r−z0 f(z) ∈ S(H 12−γ

, E).

Then the weighted Mellin transform Mγ : Tγ(R+, E)∼=−→ S(Γ 1

2−γ, E)

restricts to a topological isomorphism

Mγ : Tγ,0((0, r0), E)∼=−→ A(H 1

2−γ, E; r0)

for every r0 ∈ R+ and every γ ∈ R.

The Mellin transform in distributions. Let E be a Hilbert space.

• For γ ∈ R the space T ′γ (R+, E) consists of all continuous linear functionals

T−γ(R+) → E. Consequently, we have T ′γ (R+, E) ⊆ D′(R+, E) in view of

the density of C∞0 (R+) in T−γ(R+).

• A distribution u ∈ T ′γ (R+, E) is called regular if

〈u, ϕ〉 =

∞∫

0

u(r)ϕ(r) dr, ϕ ∈ T−γ(R+),

for some Bochner measurable function u such that r−( 12 +γ)〈logr〉−N u(r) ∈

L1(R+, E) for some N ∈ N0. In particular, we have Tγ(R+, E) ⊆T ′γ(R+, E), and more generally even L2,γ(R+, E) := rγL2(R+, E) ⊆

T ′γ(R+, E) as regular distributions.

• For every δ ∈ R the product with functions ψ ∈ C∞(R+) such thatr−δ(−r∂r)νψ(r) is majorized by some power of 〈logr〉, uniformly on R+

for every ν ∈ N0, provides an operator T ′γ(R+, E) → T ′

γ+δ(R+, E). Recall

that 〈ψu, ϕ〉 = 〈u, ψϕ〉 for ϕ ∈ T−(γ+δ)(R+).• The totally characteristic derivative (−r∂r) : D′(R+, E) −→ D′(R+, E)

restricts to T ′γ(R+, E), i. e., (−r∂r)(T ′

γ (R+, E)) ⊆ T ′γ(R+, E).


• The isomorphism Sγ : D′(R+, E) −→ D′(R+, E) from (1.1.1) restricts toan isomorphism Sγ : T ′

γ (R+, E) −→ S′(R, E). Note that we may write

〈Sγu, ϕ〉 = 〈u, S−1−γϕ〉 for ϕ ∈ S(R).

• The weighted Mellin transform Mγ extends to T ′γ (R+, E) by means of the

identity(Mγu

)(1

2− γ + iτ

)=

(√2πFSγu

)(τ), (1.1.4)

which provides an isomorphism Mγ : T ′γ (R+, E) −→ S′(Γ 1

2−γ, E). It re-

stricts to an isomorphism

Mγ : L2,γ(R+, E) −→ L2(Γ 12−γ

, E), (1.1.5)

and we have Parseval’s identity

〈u, v〉L2,γ(R+,E) =1

2π〈Mγu,Mγv〉L2(Γ 1

2−γ,E). (1.1.6)

The relations in (1.1.3) hold in the distributional sense.

Global analysis.

• In this work, we consider C∞-manifoldsX . TX denotes the tangent bundleover X , while T ∗X is the cotangent bundle. Let Vect(X) be the set ofcomplex vector bundles over X . The pull-back of a bundle F with respectto a smooth mapping g is denoted as g∗F . This is mainly employed withthe projection π : T ∗X \ 0 −→ X , where 0 is the zero section in T ∗X . LetHom(E,F ) be the bundle of homomorphisms acting in the fibres of thebundles E and F . E∗ denotes the dual bundle to E, and for vector bundlesE ∈ Vect(X) and F ∈ Vect(Y ) the external tensor product is denoted asE⊠F ∈ Vect(X×Y ).

• For a vector bundle E let C∞(X,E) denote the space of smooth sec-tions in E, and C∞

0 (X,E) is the space of smooth sections with com-pact support. Let D′(X,E) denote the distributional sections in thebundle E. With the density bundle Γ1 this space is globally given asD′(X,E) = C∞

0 (X,Γ1⊗E)′. Any choice of a smooth positive section inthe density bundle provides an isomorphism Γ1 ∼= X×C, and consequentlyD′(X,E) ∼= C∞

0 (X,E)′. Note that a Riemannian metric gives rise to acanonical positive section.

• We will be concerned mainly with closed manifolds X , i. e., X is compactand ∂X = ∅. Then we have invariantly the space L2(X,E) of measurableabsolutely square integrable sections. Any choice of a Riemannian metricand a Hermitean inner product on E induces a canonical scalar producton L2(X,E).

More generally, the Sobolev spaces Hs(X,E) of distributionalsections of smoothness s ∈ R are well-defined, where in particu-lar H0(X,E) = L2(X,E). These are hilbertizable spaces, but thechoice of an inner product is non-canonical. Hs(X,E) and H−s(X,E)are dual to each other via the sesquilinear pairing induced by the


L2(X,E)-inner product. By the Sobolev embedding theorem we haveC∞(X,E) ∼= proj-lims→∞H

s(X,E), and hence we also have an identifi-cation D′(X,E) ∼= ind-lims→−∞H

s(X,E).• Let X be closed, and let A : C∞(X,E) −→ D′(X,F ) be continuous.

Then the Schwartz kernel of A belongs to C∞(X×X,F⊠E∗) if and onlyif A extends to a continuous operator A : Hs(X,E) −→ Ht(X,F ) for alls, t ∈ R.

1.2. Finitely meromorphic Fredholm families in Ψ-algebras

Remark 1.2.1. Ψ- and Ψ∗-algebras were introduced by Gramsch [21]. These arecertain topological Frechet-algebras which share many important properties ofBanach- and C∗-algebras. We include in this section some results about the in-version of meromorphic Fredholm families taking values in Ψ-algebras which areneeded in this exposition (see also Gohberg and Sigal [19], Gramsch [20], andGramsch and Kaballo [22]).

Definition 1.2.2. Let Ψ be a subalgebra of the unital Banach-algebra B. ThenΨ is called a Ψ-algebra in B if

i) Ψ is a locally convex Frechet space with respect to the topology τ(Ψ) whichis finer than the induced topology of B.

ii) 1Ψ = 1B.iii) Ψ is “spectrally invariant” in B, i. e., for the groups of invertible elements we

have B−1 ∩ Ψ = Ψ−1.

If B is a C∗-algebra and Ψ a symmetric Ψ-algebra in B, then Ψ is called a Ψ∗-algebra in B.

Remark 1.2.3. By the left-regular representation of B in L(B) we may assumeB = L(X) with a Banach space X . In the case of Ψ∗-algebras we may assumeB = L(H) with a Hilbert space H due to the Gelfand-Neumark-Segal theorem.

The multiplication in a Ψ-algebra is jointly continuous with respect to τ(Ψ)which follows from the closed graph theorem. Note that Ψ−1 is open and conse-quently the inversion ·−1 : Ψ−1 → Ψ−1 is continuous, since (Ψ, τ(Ψ)) is Frechet.Moreover, also the ∗-operation is continuous in a Ψ∗-algebra by the closed graphtheorem. Note furthermore that a Ψ-algebra Ψ is invariant with respect to the(one-dimensional) holomorphic functional calculus.

Definition 1.2.4. Let X be a Banach space and Ψ a Ψ-algebra in L(X). LetΩ ⊆ C be an open set and D ⊆ Ω a discrete subset. A function T ∈ A(Ω \D,Ψ)is called a finitely meromorphic Fredholm family in Ω if

i) T takes values in the Fredholm operators Φ(X).ii) For p ∈ D there exists a neighbourhood U(p) ⊆ Ω such that T can be written

T (z) =−1∑

k=−N

Fk(z − p)k + T0(z), z ∈ U(p) \ p,


with finite-dimensional operators Fk ∈ F(X) and a holomorphic functionT0 ∈ A(U(p),L(X)) such that T0(U(p)) ⊆ Φ(X).

Remark 1.2.5. Note that we allow D = ∅, i. e., the case of holomorphic Fred-holm families is contained in this definition. By Cauchy’s integral formulas for theLaurent coefficients we have

Fk =1

2πi

∫

∂Uδ(p)

T (ζ)

(ζ − p)k+1dζ for k = −N, . . . ,−1.

It follows Fk ∈ Ψ, and consequently T0 ∈ A(U(p) \ p,Ψ). But from Cauchy’sintegral formula

T0(z) =1

2πi

∫

∂Uδ(p)

T0(ζ)

ζ − zdζ, z ∈ Uδ(p),

we obtain T0 ∈ A(U(p),Ψ).Finally, we want to point out that T0(p) is required to be Fredholm which

is necessary for the validity of the theorem about the inversion of meromorphicFredholm families.

Theorem 1.2.6. Let Ψ be a Ψ-algebra in L(X) with a Banach space X. Let Ω ⊆ C

be a connected domain and T ∈ A(Ω\D,Ψ) a finitely meromorphic Fredholm familyin Ω. Let z∗ ∈ Ω \D such that T (z∗) is invertible in L(X).

Then there exists a discrete set D ⊆ Ω, D ⊆ D, such that T (z) is invertible

in L(X) for z ∈ Ω \ D. Moreover, we have T−1 ∈ A(Ω \ D,Ψ) and T−1 extendsto a finitely meromorphic Fredholm family in the sense of Definition 1.2.4.

For the proof of this theorem we need some preparations. First recall thefollowing theorem on inversion of holomorphic Fredholm families.

Theorem 1.2.7. Let Ω ⊆ C be a connected domain and T ∈ A(Ω,L(X)) takingvalues in Φ(X). Let z∗ ∈ Ω such that T (z∗) is invertible in L(X).

Then there exists a discrete set D ⊆ Ω such that T (z) is invertible in L(X)for z ∈ Ω \D. Moreover, we have T−1 ∈ A(Ω \D,L(X)) and for p ∈ D we canwrite in a neighbourhood U(p):

T (z)−1 =−1∑

k=−N

Fk(z − p)k + T0(z), z ∈ U(p) \ p,

with finite-dimensional operators Fk ∈ F(X) and T0 ∈ A(U(p),L(X)), T0(U(p)) ⊆Φ(X).

Proof. Let z0 ∈ Ω such that T (z0) is not invertible in L(X). Since Ω is con-nected and T (z∗) ∈ L(X)−1 we conclude T (Ω) ⊆ Φ0(X) where Φ0(X) denotesthe Fredholm operators of index zero. Recall that the index is locally constanton Φ(X). Thus we have 0 < dimN(T (z0)) = codimR(T (z0)). By Kato’s lemma


the range of Fredholm operators is closed and thus we can find a direct decom-position X = N(T (z0)) ⊕top X1 = X2 ⊕top R(T (z0)). By choosing an isomor-phism N(T (z0)) ∼= X2 we find a finite-dimensional operator F ∈ F(X) such thatT (z0) − F ∈ L(X)−1 and consequently T (z) − F ∈ L(X)−1 for |z − z0| < εwith ε > 0 sufficiently small. With the projections P = P 2 : X → X2,N(P ) = R(T (z0)), and Q = I − P we may write for |z − z0| < ε:

T (z) = (I + PF (z))(T (z) − F ) with

F (z) := F (T (z) − F )−1 and thus

T (z) = (I + PF (z))(T (z) − F )

= (I + PF (z)Q)︸︷︷︸=: C(z)

(I + PF (z)P ) (T (z) − F )︸︷︷︸=: B(z)

.

Note that B(z) as well as C(z) are invertible for |z − z0| < ε; we can write as atriangular matrix

C(z) =

(Q 0

PF (z)Q P

)with “invertible diagonal”.

Note also that B(z), C(z) and F (z) are holomorphic for |z − z0| < ε. Thus wehave for |z − z0| < ε

T (z) = C(z) ·(Q 00 P (I + F (z))P

)·B(z).

This implies: T (z) is invertible for |z − z0| < ε if and only if P (I + F (z))P isinvertible in L(X2). Then we have with the inverse M(z) = [P (I + F (z))P ]−1 ∈L(X2):

T (z)−1 = B(z)−1 ·(Q 00 PM(z)P

)· C(z)−1. (1)

In the case dimX <∞ the assertion of the theorem is obvious due to Cramer’s rulefor the inversion of a matrix and due to the scalar analysis of meromorphic func-tions in connected domains (applied to the determinant of component-functionsof a matrix-valued function).

Thus it remains to prove the existence of z ∈ Ω, |z − z0| < ε, such that T (z)is invertible in L(X). Employing the finite-dimensional result with the functionP (I + F (z))P and inverse M(z) we then see that T (z) is invertible for 0 < |z −z0| < δ < ε and by (1) we have that T−1 is meromorphic in z0 and the Laurent-coefficients of the principal part are finite-dimensional operators. Let D := z0 ∈Ω; T (z0) is not invertible. D is a closed subset in Ω. We will show

D = ∅, i. e.,

D = ∂D. Assume that there exists a point z1 ∈

D. Since Ω is connected, we maychoose a path γ : [0, 1] → Ω, γ(0) = z1, γ(1) = z∗. Let

s := supt > 0; T (γ(τ)) is not invertible in L(X) for τ ∈ [0, t).


By assumption we have 0 < s < 1 and γ(s) ∈ ∂D ( Ω. The first part of theproof implies the existence of 0 < δ such that T (z) in invertible in L(X) for0 < |z − γ(s)| < δ. This leads to a contradiction. Thus we have D = ∂D andby the first part of the proof D is consequently discrete in Ω. This proves thatT−1 ∈ A(Ω \D,L(X)), and for p ∈ D we have in a neighbourhood U(p):

T (z)−1 =

−1∑

k=−N

Fk(z − p)k + T0(z), z ∈ U(p) \ p,

with finite-dimensional operators Fk ∈ F(X) and T0 ∈ A(U(p),L(X)), T0(U(p) \p) ⊆ Φ(X). It remains to prove T0(p) ∈ Φ(X). We can write for z ∈ U(p) \ p:

I = T (z)T (z)−1 = T (z) ·[ −1∑

k=−N

Fk(z − p)k]

+ T (z)T0(z)

= T (z)−1T (z) =[ −1∑

k=−N

Fk(z − p)k]· T (z) + T0(z)T (z).

Since the functions T0(z)T (z) and T (z)T0(z) extend holomorphically into p the

functions T (z) ·[ −1∑k=−N

Fk(z − p)k]

and[ −1∑k=−N

Fk(z − p)k]· T (z) necessarily ex-

tend also holomorphically into p. But the latter functions take values in thefinite-dimensional operators on U(p) \ p, and thus their values in p are com-pact operators. Hence T (p) inverts T0(p) modulo compact operators which showsT0(p) ∈ Φ(X).

Lemma 1.2.8. Let X be a vector space and E1, . . . , EN ⊆ X be subspaces of finitecodimension. Then E1 ∩ . . . ∩ EN ⊆ X is of finite codimension in X.

Proof. Consider the mapping J : X →N⊕j=1

X/Ej given by the canonical quotient

mappings. J is linear, and we have N(J) =N⋂j=1

Ej . Consequently

X/

N⋂

j=1

Ej ∼= R(J) ⊆N⊕

j=1

X/Ej ,

where dimN⊕j=1

X/Ej < ∞ by assumption. This implies dimX/N⋂j=1

Ej < ∞ which

shows codimN⋂j=1

Ej <∞.

Lemma 1.2.9. Let X be a Banach space and Ω ⊆ C be a connected open neigh-bourhood of 0 ∈ C. Let A−1, . . . , A−N ∈ F(X) be finite-dimensional operators. Let


H ∈ A(Ω,L(X)) such that H(z)u = 0, z ∈ Ω, for u ∈ K0 ⊆ X, where K0 is aclosed subspace of X of finite codimension. Consider the function

F (z) := I +H(z) +

−1∑

k=−N

Akzk, z ∈ Ω \ 0.

Assume that there exists a z∗ ∈ Ω \ 0 such that F (z∗) is invertible in L(X).Then there exists a δ > 0 such that F (z) is invertible for 0 < |z| < δ. Moreover,we can write for 0 < |z| < δ

F (z)−1 =

−1∑

k=−M

Fkzk + F0(z)

with finite-dimensional operators Fk ∈ F(X) and F0 ∈ A(Uδ(0),L(X)). Further-more we have (I +H)(Ω) ⊆ Φ(X) and F0(Uδ(0)) ⊆ Φ(X).

Proof. We will first prove that (I +H)(Ω) ⊆ Φ0(X), more precisely H(z) ∈ F(X)for z ∈ Ω. Since K0 ⊆ N(H(z)) we have that the canonical mapping X/K0 −→X/N(H(z) is onto. But since dimX/K0 < ∞ we conclude dimX/N(H(z)) < ∞,

i. e., H(z) ∈ F(X). Let K1 :=−1⋂

k=−N

N(Ak) and K := K0 ∩ K1. Then K is a

closed subspace of X and by Lemma 1.2.8 we have codimK < ∞. Let L ⊆ Xbe a finite-dimensional subspace such that X = K ⊕top L and P = P 2 ∈ L(X),R(P ) = L, N(P ) = K. Consequently we may write for z ∈ Ω \ 0, Q = I − P :

F (z) =

(Q QF (z)P0 PF (z)P

)=

(Q 00 PF (z)P

)·(Q QF (z)P0 P

).

Since C(z) :=

(Q QF (z)P0 P

)is invertible for all z ∈ Ω \ 0 we see that F (z)

is invertible in L(X) if and only if PF (z)P is invertible in L(L) for z ∈ Ω \ 0where dimL < ∞. With the inverse M(z) = (PF (z)P )−1 ∈ L(L) we then maywrite

F (z)−1 = C(z)−1 ·(Q 00 PM(z)P

)=

(Q −QF (z)P0 P

)·(Q 00 PM(z)P

). (1)

For z ∈ Ω \ 0 we have C(z)−1 = I − QH(z)P −−1∑

k=−N

QAkPzk, i. e., C(z)−1

is meromorphic in 0 and the Laurent coefficients of the principal part are finite-dimensional operators. Since dimL <∞ the function PF (z)P can be regarded asa holomorphic matrix-valued function on z ∈ Ω \ 0 which is meromorphic in 0.The determinant of the component functions is consequently a holomorphic scalarfunction which is meromorphic in 0. Since F (z∗) is invertible in L(X), we have thatPF (z∗)P is invertible in L(L), i. e., the determinant of the component functions isa meromorphic scalar function in Ω which is not identically zero. From Cramer’srule for the inversion of a matrix we now get that the functionM(z) = (PF (z)P )−1

is a meromorphic L(L)-valued function. Note that Ω is assumed to be connected


and thus the scalar meromorphic functions in Ω form a field. In particular, we

see that there exists a δ > 0 such that

(Q 00 PF (z)P

)is invertible in L(X) for

0 < |z| < δ (and consequently also F (z)), and we may write for 0 < |z| < δ

(Q 00 PF (z)P

)−1

=

(Q 00 PM(z)P

)= Z0(z) +

−1∑

k=−M

Zkzk

with finite-dimensional operators Zk ∈ F(X) and Z0 ∈ A(Uδ(0),L(X)). From theidentity (1) we now get that F (z)−1, z ∈ Uδ(0)\0, is a product of two meromor-phic functions (meromorphic in 0), whose Laurent coefficients of the principal partsare finite-dimensional operators. This proves that for 0 < |z| < δ we may write

F (z)−1 =−1∑

k=−M

Fkzk + F0(z) with finite-dimensional operators Fk ∈ F(X) and

F0 ∈ A(Uδ(0),L(X)), F0(Uδ(0) \ 0) ⊆ Φ(X). It remains to prove F0(0) ∈ Φ(X).Let [·] : L(X) −→ L(X)/K(X) be the canonical quotient mapping, where K(X)denotes the ideal of compact operators. For 0 < |z| < δ we have

[I] = [F (z)F (z)−1] = [F (z)][F (z)−1] = [I +H(z)][F0(z)]

= [F (z)−1F (z)] = [F (z)−1][F (z)] = [F0(z)][I +H(z)],

and consequently [I] = [I +H(0)][F0(0)] = [F0(0)][I +H(0)], since the functionsI+H and F0 are holomorphic in 0. This shows that I+H(0) inverts F0(0) modulocompact operators, i. e., F0(0) ∈ Φ(X).

Proof of Theorem 1.2.6. We will first prove the assertion in the case Ψ = L(X).Since D is discrete in Ω we have that Ω \D is a connected domain. Hence we

may apply Theorem 1.2.7 on the inversion of holomorphic Fredholm families to thefunction T ∈ A(Ω\D,L(X)). It follows the existence of a discrete set D′ ⊆ (Ω\D)such that T (z) is invertible in L(X) for z ∈ (Ω \D) \D′ and the inverse T−1 is ameromorphic Fredholm family in (Ω \D) \D′ in the sense of Definition 1.2.4. Itremains to prove that no point p ∈ D is an accumulation point of D′ and that T−1

extends meromorphically into p ∈ D, i. e., there exists a neighbourhood U(p) of p

such that we can write for z ∈ U(p)\ p: T−1(z) =−1∑

k=−M

Fk(z−p)k+ F0(z) with

Fk ∈ F(X) and F0 ∈ A(U(p),L(X)), F0(U(p)) ⊆ Φ(X). Let p ∈ D. By assumption

we find ε > 0 such that for 0 < |z−p| < ε we have T (z) =−1∑

k=−N

Fk(z−p)k+T0(z)

with Fk ∈ F(X) and T0 ∈ A(Uε(p),L(X)), T0(Uε(p)) ⊆ Φ(X). Since D′ is discretein Ω \ D there exists a z ∈ Uε(p) \ p such that T (z) is invertible in L(X).Consequently T0(Uε(p)) ⊆ Φ0(X) where Φ0(X) denotes the subset of all Fredholmoperators with index equal to zero. Recall that the index is locally constant onΦ(X) and that Uε(p) is connected. Let F0 ∈ F(X) be a finite-dimensional operatorsuch that T0(p) − F0 is invertible in L(X). Thus T0(z) − F0 is invertible in L(X)


for all z in a small neighbourhood of p. Without loss of generality we may assumethat T0(z) − F0 is invertible for z ∈ Uε(p). For 0 < |z − p| < ε we can write

(T0(z) − F0)−1T (z) = (T0(z) − F0)

−1[ −1∑

k=−N

Fk(z − p)k]+ (T0(z) − F0)

−1T0(z)

= I + (T0(z) − F0)−1

[ 0∑

k=−N

Fk(z − p)k]

=: F (z).

We have (T0(z) − F0)−1

[ 0∑k=−N

Fk(z − p)k]

= H(z) +−1∑

k=−N

Ak(z − p)k with

finite-dimensional operators Ak ∈ F(X) and H ∈ A(Uε(p),L(X)). Set K0 :=( 0⋂k=−N

N(Fk))∩

( −1⋂k=−N

N(Ak)). Then K0 is a closed subspace of X and by Lemma

1.2.8 we have codimK0 <∞. This implies H(z)u = 0 for u ∈ K0, z ∈ Uε(p) \ p,but from Cauchy’s integral formula

H(p) =1

2πi

∫

∂U ε2(p)

H(ζ)

ζ − pdζ

we also obtain H(p)u = 0 for u ∈ K0. Thus we may apply Lemma 1.2.9 to thefunction F (z) in Uε(p) \ p which shows the existence of 0 < δ < ε, such thatF (z) = (T0(z) − F0)

−1T (z) is invertible for 0 < |z − p| < δ. Consequently T (z) isinvertible in L(X) for 0 < |z−p| < δ which implies that p is no accumulation point

of D′, i. e., D := D∪D′ is discrete in Ω. Moreover, we can write for 0 < |z−p| < δ:

F (z)−1 = T (z)−1 · (T0(z) − F0) =

−1∑

k=−M

Ak(z − p)k + A0(z)

with finite-dimensional operators Ak ∈ F(X) and A0 ∈ A(Uδ(p),L(X)), and hence

T (z)−1 =[ −1∑

k=−M

Ak(z − p)k]· (T0(z) − F0)

−1 + A0(z) · (T0(z) − F0)−1

=−1∑

k=−M

Fk(z − p)k + F0(z)

with Fk ∈ F(X) and F0 ∈ A(Uδ(p),L(X)). Since F0(Uδ(p) \ p) ⊆ Φ(X) it

remains to prove that F0(p) ∈ Φ(X).Let [·] : L(X) → L(X)/K(X) be the canonical quotient mapping, where

K(X) denotes the ideal of compact operators. For 0 < |z − p| < δ we have

[I] = [T (z) · T (z)−1] = [T (z)][T (z)−1] = [T0(z)][F0(z)]

= [T (z)−1 · T (z)] = [T (z)−1][T (z)] = [F0(z)][T0(z)]


and consequently [I] = [T0(p)][F0(p)] = [F0(p)][T0(p)] since the functions F0 and

T0 are holomorphic in p. This shows F0(p) ∈ Φ(X) and finishes the proof ofTheorem 1.2.6 in the case Ψ = L(X).

In the general case we may first apply the result for L(X). It remains to show

that T (z)−1 ∈ Ψ for z ∈ Ω \ D and that T−1 ∈ A(Ω \ D,Ψ). But by Definition1.2.2 of Ψ-algebras we have T (z)−1 ∈ Ψ, and since the inversion −1 : Ψ−1 → Ψ−1

is continuous we obtain the holomorphy of T−1 in Ω \ D from the holomorphy ofT (as Ψ - valued functions). This completes the proof of Theorem 1.2.6.

Example 1.2.10. We conclude this section with an example where the validity ofTheorem 1.2.6 is violated due to the holomorphic part of a meromorphic functionevaluated at a pole not being a Fredholm operator.

Let X be a Banach space and P = P 2 ∈ L(X) be a non-trivial finite-dimensional projection. Assume dimX = ∞. Consider the function T ∈ A(C \0,L(X)) given by T (z) := zI − 1

zP . We have T (C \ 0) ⊆ Φ0(X), and T (z) is

invertible in L(X) for |z| > ‖P‖ 12 . For |z| large we may write

T (z)−1 =1

z(I − 1

z2P )−1 =

1

z·

∞∑

k=0

1

z2kP k

=1

z·[I +

( ∞∑

k=1

1

z2k

)P

]= (I − P )

1

z+

z

z2 − 1P.

Since T (z) is invertible for z ∈ C \ 0,−1,+1 we conclude from uniqueness ofanalytic continuation that

T (z)−1 = (I − P )1

z+

z

z2 − 1P for z ∈ C \ 0,−1,+1.

Hence T−1 is meromorphic in 0 but the residue is (I − P ) /∈ F(X).

1.3. Volterra integral operators

Remark 1.3.1. In this section we discuss integral operators with operator-valuedkernel functions that are supported on one side of the diagonal. The theory ofoperators of such kind is classical, and they arise, e. g., in the study of (Volterra)integral equations. From our point of view the main property of these operatorsis, that under some natural assumptions they turn out to be quasinilpotent, i. e.,their spectrum consists of zero only. This observation will be employed later in theanalysis of remainders of the parametrix construction to parabolic pseudodifferen-tial operators, and it is crucial for the proof of the invertibility of these operatorswithin the calculus.

Remark 1.3.2. Let E, E and E be Hilbert spaces, and let I ⊆ R be an interval.


a) With a kernel function k ∈ L2(I×I,L(E, E)) we associate an operator Tk ∈L(L2(I, E), L2(I, E)) via

(Tku

)(t) :=

∫

I

k(t, t′)u(t′) dt′

for u ∈ L2(I, E). The mapping

T : L2(I×I,L(E, E)) ∋ k 7−→ Tk ∈ L(L2(I, E), L2(I, E))

is one-to-one and bounded with norm ‖T ‖ = 1.

In particular, T is an isomorphism of L2(I×I,L(E, E)) onto its range in

L(L2(I, E), L2(I, E)), and via T we transfer the kernel topology to the operator

space. Thus we obtain a Banach subspace of L(L2(I, E), L2(I, E)) endowedwith a finer topology.

b) Let k1 ∈ L2(I×I,L(E, E)) and k2 ∈ L2(I×I,L(E, E)). Then the composition

Tk1Tk2 ∈ L(L2(I, E), L2(I, E)) equals Tk1k2 with the function

(k1k2

)(t, t′) :=

∫

I

k1(t, s)k2(s, t′) ds ∈ L2(I×I,L(E, E)).

The mapping

: L2(I×I,L(E, E))×L2(I×I,L(E, E)) −→ L2(I×I,L(E, E))

is bilinear and continuous; more precisely we have ‖k1k2‖L2 ≤ ‖k1‖L2‖k2‖L2.

Definition 1.3.3. Let k ∈ L2(I×I,L(E, E)). The operator Tk is called a Volterraintegral operator provided that one of the following equivalent conditions is ful-filled:

i) For every t0 ∈ I we have(Tku

)(t) ≡ 0 for t > t0 for all u ∈ L2(I, E) such

that u(t) ≡ 0 for t > t0.

ii) For every u ∈ L2(I, E) and every v ∈ L2(I, E) such that supp(u) < supp(v)we have 〈Tku, v〉L2(I,E) = 0.

iii) k(t, t′) ≡ 0 for t > t′.

A kernel k satisfying iii) is called Volterra integral kernel.

Proposition 1.3.4. a) The space of Volterra integral kernels is a closed subspace

of L2(I×I,L(E, E)).

b) Let k1 ∈ L2(I×I,L(E, E)) and k2 ∈ L2(I×I,L(E, E)) be Volterra integral

kernels. Then also k1k2 ∈ L2(I×I,L(E, E)) is a Volterra integral kernel. Ifk1 and k2 are continuous then k1k2 is continuous.

On the level of operators this means that the space of Volterra integral oper-ators is a closed subspace of all integral operators with L2-kernel functions, and itis closed with respect to taking compositions.

Proof. These assertions are obvious. For the continuity of k1k2 in b) let us notethe following:


To every point (t0, t′0) ∈ I×I there exists a neighbourhood U(t0, t

′0) ⊆ I×I

and a compact subinterval J ⊆ I such that

(k1k2

)(t, t′) =

∫

J

k1(t, s)k2(s, t′) ds

for (t, t′) ∈ U(t0, t′0). Thus the continuity follows from the continuity of k1 and k2

and Lebesgue’s dominated convergence theorem.

Lemma 1.3.5. Let k ∈ L2(I×I,L(E)) be a continuous Volterra integral kernel.Moreover, let g, h ∈ C(I) be everywhere positive functions, and assume that

C := supg(t)h(t′)‖k(t, t′)‖L(E); (t, t′) ∈ I×I <∞.

For short we writek(N) := k . . . k︸︷︷︸

N

∈ L2(I×I,L(E))

for N ∈ N. Then k(N) is a continuous Volterra integral kernel, and for t′ ≥ t wehave the pointwise estimate

g(t)h(t′)‖k(N)(t, t′)‖L(E) ≤ CN

1

(N − 1)!

( t′∫

t

1

g(s)h(s)ds

)N−1

.

Proof. k(N) is a continuous Volterra integral kernel by Proposition 1.3.4. It remains

to prove the pointwise estimate. Let F ∈ C1(I) such that F ′ = 1gh . We proceed

by induction: For N = 1 the estimate is true by assumption. Now assume it holdsfor some N ∈ N. Then we have for t′ ≥ t:

g(t)h(t′)‖k(N+1)(t, t′)‖L(E) = g(t)h(t′)‖

(kk(N)

)(t, t′)‖L(E)

= g(t)h(t′)∥∥∥t′∫

t

k(t, s)k(N)(s, t′) ds

∥∥∥L(E)

=∥∥∥t′∫

t

1

g(s)h(s)

(g(t)h(s)k(t, s)

)(g(s)h(t′)k(N)(s, t

′))ds

∥∥∥L(E)

≤ CN+1 1

(N − 1)!

t′∫

t

F ′(s)(F (t′) − F (s)

)N−1ds

= CN+1 1

N !

(F (t′) − F (t)

)N.

This finishes the proof of the lemma.

Theorem 1.3.6. Let g, h ∈ C(I) be everywhere positive functions with∫

I

1

g(s)2ds <∞ and

∫

I

1

h(s)2ds <∞.


Moreover, let k ∈ C(I×I,L(E)) such that k(t, t′) ≡ 0 for t > t′, and

supg(t)h(t′)‖k(t, t′)‖L(E); (t, t′) ∈ I×I <∞.

Then k ∈ L2(I×I,L(E)) is a continuous Volterra integral kernel, and the Volterraintegral operator Tk ∈ L(L2(I,L(E))) is quasinilpotent. For 0 6= λ ∈ C we have

(λId − Tk

)−1=

1

λId− Tk′

with a Volterra integral operator Tk′ .

Proof. Clearly, k ∈ L2(I×I,L(E)) is a continuous Volterra integral kernel. Let

C := supg(t)h(t′)‖k(t, t′)‖L(E); (t, t′) ∈ I×I.Then we have for t′ ≥ t in the notation from Lemma 1.3.5:

g(t)h(t′)‖k(N)(t, t′)‖L(E) ≤ CN

1

(N − 1)!

( t′∫

t

1

g(s)h(s)ds

)N−1

≤ CN1

(N − 1)!

∥∥∥1

g

∥∥∥N−1

L2(I)

∥∥∥1

h

∥∥∥N−1

L2(I),

and consequently

‖k(N)‖L2(I×I,L(E)) ≤

(C

∥∥∥ 1g

∥∥∥L2(I)

∥∥∥ 1h

∥∥∥L2(I)

)N

(N − 1)!.

This shows that

‖TNk ‖1N

L(L2(I,L(E))) ≤ ‖k(N)‖1N

L2(I×I,L(E)) −→N→∞

0,

i. e., Tk is quasinilpotent. Moreover, for λ 6= 0 the series

k′ := −∞∑

N=1

1

λN+1k(N) ∈ L2(I×I,L(E))

converges and defines a Volterra integral kernel, and we have(λId − Tk

)−1=

1

λId − Tk′ .

Some notes on abstract kernels.

Remark 1.3.7. In many situations the residual elements of a pseudodifferential cal-culus are characterized by their mapping properties in a scale of suitable functionspaces. In algebras consisting of Volterra operators we are interested to invert theremainders of the parametrix construction of parabolic elements within the calcu-lus, where Theorem 1.3.6 serves as the key for the proof. In order to be able toapply this result we are in need to obtain information about the Schwartz kernelsof the residual elements from their mapping properties.

To this end recall the following facts (cf., e. g., Jarchow [29]):


a) Let E and F be (Hausdorff) locally convex spaces. Then E⊗F is realized withinEεF := Le(E′

c, F ) vian∑

i=1

ei⊗fi 7−→(e′ 7→

n∑

i=1

〈e′, e〉E′,E fi

).

Here the subscript c denotes the topology of uniform convergence on precom-pact subsets in E, while the subscript e denotes the topology of uniform con-vergence on equicontinuous subsets in E′. The induced topology of EεF onE⊗F is the ε-topology, i. e., we obtain the injective tensor product E⊗εF ofthe spaces E and F .

b) EεF is complete if and only if F is complete.c) If E and F are complete then EεF and FεE are topologically isomorphic via

transposition, i. e.,EεF ∋ G 7−→ Gt ∈ FεE.

Passing to completions shows that this isomorphism induces in any case atopological isomorphism E⊗εF ∼= F⊗εE.

d) If F has the approximation property, which in particular holds for hilbertizableand consequently for nuclear spaces F , then E⊗F is dense in EεF .

e) From b) and d) we conclude that E⊗εF = EεF if F is complete and has theapproximation property.

f) If E or F is nuclear we have E⊗πF = E⊗εF . In particular, if F is completeand has the approximation property, we have E⊗πF = E⊗εF = EεF .

g) Let E and F be Frechet spaces and assume that E is nuclear. Then we haveLβ(E′

β , F ) ∼= E⊗πF in the canonical way. Here the subscript β indicates thatthe spaces are endowed with the strong topology.

Remark 1.3.8. Let E0, E,E1 be a Hilbert triple. Then the inner product in Einduces a canonical (antilinear) isomorphism E1

∼= E′0 via

E1 ∋ e1 7−→ 〈·, e1〉E ∈ E′0.

Let E be another Hilbert space. Via this isomorphism we may identify the nuclearoperators E0 −→ E as

ℓ1(E0, E) = E′0⊗πE ∼= E1⊗πE,

i. e., G ∈ ℓ1(E0, E) if and only if there exist sequences (λj) ∈ ℓ1 and (xj) ⊆ E1,

(ej) ⊆ E tending to zero such that

G =

∞∑

j=1

λj〈·, xj〉E ej .

The tensor product representations in Proposition 1.3.9 below are to be understoodin this sense.

Proposition 1.3.9. Let E0, E,E1 and E0, E, E1 be Hilbert triples, and let F

and F be nuclear Frechet spaces such that F → E1 and F → E0. Moreover, letG ∈ L(E0, E0) be given.


a) We have G(E0) ⊆ F if and only if

G ∈ E1⊗πF → ℓ1(E0, E0).

b) G(E0) ⊆ F and G∗(E1) ⊆ F if and only if

G ∈(E1⊗πF

)∩

(F ⊗πE0

)→ ℓ1(E0, E0).

c) Let E0, E, E1 be another Hilbert triple, and assume that G = AB∗ with A ∈L(E0, E0) and B ∈ L(E1, E1) such that A(E0) ⊆ F and B(E1) ⊆ F . Then wehave

G ∈ F ⊗πF → ℓ1(E0, E0).

Proof. For the proof of a) note first that G(E0) ⊆ F if and only if G ∈ L(E0, F )

by the closed graph theorem. Clearly, every element in E1⊗πF ∼= E′0⊗πF induces

an operator in L(E0, F ). Now assume that G ∈ L(E0, F ) is given. Then we have

by the nuclearity of F that

Gt ∈ L((F ′)β , E′0) = L((F ′)c, E

′0) = F εE′

0,

and thus by Remark 1.3.7 (and 1.3.8)

G =(Gt

)t ∈ E′0εF = E′

0⊗πF ∼= E1⊗πF .

Assume that G fulfills the mapping properties in b). From a) we obtain that

G ∈ E1⊗πF and G∗ ∈ E0⊗πF ∼= E′1⊗πF . Since G = (G∗)∗ we conclude that

G ∈ F ⊗πE0, i. e.,

G ∈(E1⊗πF

)∩

(F ⊗πE0

)

as desired. The converse assertion in b) is immediate.

For the proof of c) note first that by a) we have A ∈ E1⊗πF . Consequently,

there are sequences (λj) ∈ ℓ1 and (xj) ⊆ E1, (fj) ⊆ F tending to zero such that

A(e) =

∞∑

j=1

λj〈e, xj〉E fj

for e ∈ E0. Thus we may write for e ∈ E0:

A(B∗e) =

∞∑

j=1

λj〈B∗e, xj〉E fj =

∞∑

j=1

λj〈e,B(xj)〉E fj .

Since B ∈ L(E1, F ) we conclude that the sequence (B(xj)) ⊆ F converges to zero,

i. e., G = AB∗ ∈ F ⊗πF as asserted.


Chapter 2. Abstract Volterra pseudodifferential calculus

2.1. Anisotropic parameter-dependent symbols

Definition 2.1.1. Let ℓ ∈ N be a given anistropy.

a) For (ξ, λ) ∈ Rn × Rq define

|ξ, λ|ℓ := (|ξ|2ℓ + |λ|2) 12ℓ ,

〈ξ, λ〉ℓ := (1 + |ξ|2ℓ + |λ|2) 12ℓ ,

where | · | denotes the Euclidean norm.

b) For a multi-index β = (α, α′) ∈ Nn+q0 let

|β|ℓ := |α| + ℓ · |α′|,

where |·| denotes the usual length of a multi-index as the sum of its components.

Lemma 2.1.2. There exists a constant c > 0 such that for all s ∈ R and ξ1, ξ2 ∈Rn, λ1, λ2 ∈ Rq the following inequality is fulfilled (Peetre’s inequality):

〈ξ1 + ξ2, λ1 + λ2〉sℓ ≤ c|s| 〈ξ1, λ1〉|s|ℓ 〈ξ2, λ2〉sℓ . (2.1.1)

Morover, we can compare the regularized “anisotropic distance” 〈·, ·〉ℓ with the“isotropic distance”, i. e., there exist constants c1, c2 > 0 such that

c1 〈ξ, λ〉ℓ ≤ 〈ξ, λ〉 ≤ c2 〈ξ, λ〉ℓℓ . (2.1.2)

Definition 2.1.3. Let E and E be Hilbert spaces. For µ ∈ R we define

Sµ;ℓ(Rn × Rq;E, E) := a ∈ C∞(Rn × Rq,L(E, E));

pk(a) := sup(ξ,λ)∈R

n×Rq

|β|ℓ≤k

‖∂β(ξ,λ)a(ξ, λ)‖ 〈ξ, λ〉−µ+|β|ℓℓ <∞ for all k ∈ N0.

This is a Frechet space with the topology induced by the seminorm-system pk; k ∈N0. Define

S−∞(Rn × Rq;E, E) :=⋂

µ∈R

Sµ;ℓ(Rn × Rq;E, E).

By (2.1.2) this space does not depend on ℓ ∈ N, and we have S−∞(Rn×Rq;E, E) =

S(Rn × Rq,L(E, E)). Moreover, for µ ∈ R the spaces of x- (resp. x′-) and (x, x′)-dependent symbols are defined as

Sµ;ℓ(Rn × Rn × Rq;E, E) := C∞b (Rn, Sµ;ℓ(Rn × Rq;E, E)),

Sµ;ℓ(Rn × Rn × Rn × Rq;E, E) := C∞b (Rn × Rn, Sµ;ℓ(Rn × Rq;E, E)).

Analogously, we obtain the spaces of order −∞. If E = E = C we suppress theHilbert spaces from the notation.


Let Ejj∈N and Ejj∈N be scales of Hilbert spaces such that Ej → Ej+1

and Ej+1 → Ej for j ∈ N. Define

Sµ;ℓ(Rn × Rq; ind-limj∈N

Ej , proj-limk∈N

Ek) :=⋂

j,k∈N

Sµ;ℓ(Rn × Rq;Ej , Ek)

with the natural Frechet topologies induced. The spaces of order −∞ are defined inan analogous manner, as well as the symbol spaces with dependence on x, x′ ∈ Rn.With this notion the case of single Hilbert spaces E and E corresponds to theconstant scales.

Definition 2.1.4. Let E and E be Hilbert spaces. A function f : (Rn × Rq) \0 → L(E, E) is called (anisotropic) homogeneous of degree µ ∈ R, if for (ξ, λ) ∈(Rn × Rq) \ 0 and > 0

f(ξ, ℓλ) = µf(ξ, λ). (2.1.3)

A function f : Rn × Rq → L(E, E) is called (anisotropic) homogeneous of degreeµ ∈ R for large (ξ, λ), if for (ξ, λ) ∈ Rn × Rq with |(ξ, λ)| sufficiently large and ≥ 1

f(ξ, ℓλ) = µf(ξ, λ). (2.1.4)

In this work, homogeneity always is meant in this anisotropic sense.

Remark 2.1.5. Let a ∈ C∞(Rn × Rq,L(E, E)) be homogeneous of degree µ ∈ R

for large (ξ, λ). Then a ∈ Sµ;ℓ(Rn × Rq;E, E).

Asymptotic expansion.

Definition 2.1.6. Let Ej and Ej be scales of Hilbert spaces in the sense ofDefinition 2.1.3. For short, we set

E := ind-limj∈N

Ej and E := proj-limj∈N

Ej .

Let (µk) ⊆ R be a sequence of reals such that µk −→k→∞

−∞ and µ := maxk∈N

µk.

Moreover, let ak ∈ Sµk;ℓ(Rn×Rn×Rq; E , E). A symbol a ∈ Sµ;ℓ(Rn×Rn×Rq; E , E)is called the asymptotic expansion of the ak, if for every R ∈ R there is a k0 ∈ N

such that for k > k0

a−k∑

j=1

aj ∈ SR;ℓ(Rn × Rn × Rq; E , E).

The symbol a is uniquely determined modulo S−∞(Rn × Rn × Rq; E , E).

For short we write a ∼∞∑j=1

aj .

Lemma 2.1.7. Let Ej and Ej be scales of Hilbert spaces, and E and E as inDefinition 2.1.6. Let (µk) ⊆ R such that µk > µk+1 −→

k→∞−∞. Furthermore, for

each k ∈ N let (Akj )j∈N ⊆ Sµk;ℓ(Rn × Rq; E , E) be a countable system of bounded


sets. Let χ ∈ C∞(Rn × Rq) be a 0-excision function. Then there is a sequence(ci) ⊆ R+ with ci < ci+1 −→

i→∞∞ such that for each k ∈ N

∞∑

i=k

supa∈Aij

p(χ( ξ

di,λ

dℓi

)a(ξ, λ)

)<∞ (2.1.5)

for all continuous seminorms p on Sµk;ℓ(Rn × Rq; E , E) and every j ∈ N, and forall sequences (di) ⊆ R+ with di ≥ ci.

Proof. The proof of this lemma is a variant of the standard Borel-argument.

Theorem 2.1.8. Let Ej and Ej be scales of Hilbert spaces, and E and E as inDefinition 2.1.6. Let (µk) ⊆ R such that µk −→

k→∞−∞ and µ := max

k∈N

µk. Moreover,

let ak ∈ Sµk;ℓ(Rn ×Rn×Rq; E , E). Then there exists a ∈ Sµ;ℓ(Rn×Rn ×Rq; E , E)

such that a ∼∞∑j=1

aj, and a is uniquely determined modulo S−∞(Rn×Rn×Rq; E , E).

Proof. Without loss of generality we may assume that µk > µk+1 −→k→∞

−∞. For

k, j ∈ N let

Akj :=∂αx ak(x); x ∈ Rn, |α| ≤ j

.

Then Akj ⊆ Sµk;ℓ(Rn×Rq; E , E) is bounded. Let χ ∈ C∞(Rn×Rq) be a 0-excisionfunction. Now apply Lemma 2.1.7. With a suitable sequence (ci) ⊆ R+ formula(2.1.5) becomes

∞∑

i=k

supp(χ( ξci,λ

cℓi

)(∂αx ai(x))(ξ, λ)

); x ∈ Rn, |α| ≤ j

<∞

for all continuous seminorms p on Sµk;ℓ(Rn×Rq; E , E), which shows that for everyk ∈ N the sum

∞∑

i=k

χ( ·ci,·cℓi

)ai

is unconditionally convergent in Sµk;ℓ(Rn × Rn × Rq; E , E). The assertion of thetheorem follows with

a :=

∞∑

i=1

χ( ·ci,·cℓi

)ai ∈ Sµ1;ℓ(Rn × Rn × Rq; E , E).

Classical symbols.

Definition 2.1.9. Let E and E be Hilbert spaces. For µ ∈ R define

Sµ;ℓcl (Rn × Rq;E, E) :=

a ∈ Sµ;ℓ(Rn × Rq;E, E); a ∼

∞∑

k=0

χa(µ−k)

,


where χ ∈ C∞(Rn × Rq) is a 0-excision function, and a(µ−k) ∈ C∞((Rn × Rq) \0,L(E, E)) are (anisotropic) homogeneous functions of degree µ−k, the so calledhomogeneous components of a.

Remark 2.1.10. By 2.1.5 the space Sµ;ℓcl (Rn × Rq;E, E) is well-defined.

The homogeneous components of a ∈ Sµ;ℓcl (Rn × Rq;E, E) are uniquely de-

termined by a. They can iteratively be recovered from the relation

1

µ−k

(a(ξ, ℓλ) −

k−1∑

j=0

a(µ−j)(ξ, ℓλ)

)−→→∞

a(µ−k)(ξ, λ) (2.1.6)

with convergence in L(E, E), which holds locally uniformly for 0 6= (ξ, λ) ∈ Rn ×Rq.

Note that Sµ;ℓcl (Rn×Rq;E, E) is a Frechet space with respect to the projective

topology of the mappings

Sµ;ℓcl (Rn × Rq;E, E) ∋ a 7−→

a−

k−1∑j=0

χa(µ−j) ∈ Sµ−k;ℓ(Rn × Rq;E, E)

a(µ−k) ∈ C∞((Rn × Rq) \ 0,L(E, E))

for k ∈ N0.The spaces of x- (resp. x′-) and (x, x′)-dependent classical symbols are defined

as

Sµ;ℓcl (Rn × Rn × Rq;E, E) := C∞

b (Rn, Sµ;ℓcl (Rn × Rq;E, E)),

Sµ;ℓcl (Rn × Rn × Rn × Rq;E, E) := C∞

b (Rn × Rn, Sµ;ℓcl (Rn × Rq;E, E)).

Clearly, the spaces of classical symbols are closed with respect to takingasymptotic expansions if the sequence (µk)k∈N0 of orders is given as µk := µ − kfor some µ ∈ R.

Remark 2.1.11. The notions of parameter-dependent symbols are analogous if theparameter-space Rq is replaced by a conical subset ∅ 6= Λ ⊆ Rq, which is theclosure of its interior. There only arise notational modifications. In this work, wewill mainly make use of parameter-dependent symbols and operators with theparameter running over R or over a half-plane H ⊆ C ∼= R2.

2.2. Anisotropic parameter-dependent operators

Definition 2.2.1. Let E and E be Hilbert spaces, and let µ ∈ R. With a double-symbol a ∈ Sµ;ℓ(Rn×Rn×Rn×Rq;E, E) we associate a family of pseudodifferential

operators opx(a)(λ) ∈ L(S(Rn, E),S(Rn, E)) for λ ∈ Rq by means of the following


oscillatory integral:

(opx(a)(λ)u)(x) : =

∫

Rn

∫

Rn

ei(x−x′)ξa(x, x′, ξ, λ)u(x′) dx′ dξ

=

∫

Rn

∫

Rn

e−ix′ξa(x, x+ x′, ξ, λ)u(x+ x′) dx′ dξ

where as usually dξ := (2π)−ndξ.The space of these operators is denoted by

Lµ;ℓ(cl)(R

n; Rq;E, E) := opx(a)(λ); a ∈ Sµ;ℓ(cl)(R

n × Rn × Rn × Rq;E, E).

In the case of E = E = C the Hilbert spaces are suppressed from the notation.

Elements of the calculus.

Theorem 2.2.2. Let a ∈ Sµ;ℓ(Rn×Rn×Rn×Rq;E, E). Then there exist unique

left- and right-symbols aL(x, ξ, λ), aR(x′, ξ, λ) ∈ Sµ;ℓ(Rn × Rn × Rq;E, E) suchthat opx(a)(λ) = opx(aL)(λ) = opx(aR)(λ) as operators on S(Rn, E).

These symbols are given by means of the following oscillatory integrals:

aL(x, ξ, λ) =

∫∫e−iyηa(x, y + x, ξ + η, λ) dy dη,

aR(x′, ξ, λ) =

∫∫e−iyηa(x′ + y, x′, ξ − η, λ) dy dη.

The mappings a 7→ aL and a 7→ aR are continuous. Moreover, we have the asymp-totic expansions

aL(x, ξ, λ) ∼∑

α∈Nn0

1

α!∂αξ D

αx′a(x, x′, ξ, λ)|x′=x,

aR(x′, ξ, λ) ∼∑

α∈Nn0

1

α!(−1)|α|∂αξ D

αxa(x, x

′, ξ, λ)|x=x′ .

If a is classical, so are aL and aR, and the mappings a 7→ aL and a 7→ aRare continuous with respect to the (stronger) topology of classical symbols.

Remark 2.2.3. By Theorem 2.2.2 the mapping opx provides an isomorphism be-tween the space of x-dependent symbols (“left-symbols”) and pseudodifferentialoperators:

Sµ;ℓ(cl)(R

n × Rn × Rq;E, E)opx−→∼=

Lµ;ℓ(cl)(R

n; Rq;E, E).

Via opx we transfer the topology, which makes Lµ;ℓ(cl)(R

n; Rq;E, E) a Frechet space.


Moreover, we have the space of parameter-dependent operators of order −∞which is independent of ℓ ∈ N:

L−∞(Rn; Rq;E, E) =⋂

µ∈R

Lµ;ℓ(Rn; Rq;E, E) = S(Rq , L−∞(Rn;E, E))

= opx(a)(λ); a ∈ S−∞(Rn × Rn × Rq;E, E).

Theorem 2.2.4. a) Let E, E and E be Hilbert spaces. Let A(λ) = opx(a)(λ) ∈Lµ;ℓ

(cl)(Rn; Rq; E, E) and B(λ) = opx(b)(λ) ∈ Lµ

′;ℓ(cl) (R

n; Rq;E, E) with a ∈Sµ;ℓ

(cl)(Rn×Rn×Rq; E, E) and b ∈ Sµ

′;ℓ(cl) (Rn×Rn×Rq;E, E). Then the compo-

sition as operators on S(Rn, E) belongs to Lµ+µ′;ℓ(cl) (Rn; Rq;E, E).

More precisely, we have A(λ)B(λ) = C(λ) = opx(a#b)(λ) with the symbol

a#b ∈ Sµ+µ′;ℓ(cl) (Rn × Rn × Rq;E, E) given by the oscillatory integral formula

a#b(x, ξ, λ) =

∫∫e−iyηa(x, ξ + η, λ)b(x+ y, ξ, λ) dy dη. (2.2.1)

Moreover, the following asymptotic expansion holds for a#b:

a#b ∼∑

α∈Nn0

1

α!(∂αξ a)(D

αx b). (2.2.2)

The mapping (a, b) 7→ a#b is bilinear and continuous. The symbol a#b is calledthe Leibniz-product of a and b.

b) Let E0, E,E1 and E0, E, E1 be Hilbert triples, and A(λ) = opx(a)(λ) ∈Lµ;ℓ

(cl)(Rn; Rq;E0, E0) with the symbol a ∈ Sµ;ℓ

(cl)(Rn×Rn×Rq;E0, E0). Then the

formal adjoint operators belong to Lµ;ℓ(cl)(R

n; Rq; E1, E1), i. e., for u ∈ S(Rn, E0)

and v ∈ S(Rn, E1) we have 〈A(λ)u, v〉L2(Rn,E) = 〈u,A(λ)(∗)v〉L2(Rn,E) with

A(λ)(∗) = opx(a(∗))(λ), where a(∗) ∈ Sµ;ℓ

(cl)(Rn × Rn × Rq; E1, E1) is given by

means of the oscillatory integral

a(∗)(x, ξ, λ) =

∫∫e−iyηa∗(x+ y, ξ + η, λ) dy dη, (2.2.3)

and the following asymptotic expansion is valid:

a(∗) ∼∑

α∈Nn0

1

α!∂αξ D

αxa

∗. (2.2.4)

The mapping a 7→ a(∗) is antilinear and continuous. The symbol a(∗) is calledthe adjoint symbol to a.

Proof. To prove a), we associate to the operatorsB(λ) the right-symbol bR(x′, ξ, λ)according to Theorem 2.2.2. Then the composition A(λ)B(λ) has the double-

symbol c(x, x′, ξ, λ) = a(x, ξ, λ)bR(x′, ξ, λ) ∈ Sµ+µ′;ℓ(cl) (Rn × Rn × Rn × Rq;E, E).

Employing again 2.2.2, we obtain a#b as the corresponding left-symbol associatedto c. This also implies the continuity of the bilinear mapping (a, b) 7→ a#b.


Assertion b) follows directly from Theorem 2.2.2, noting that(a(x′, ξ, λ)

)∗is

the right-symbol for A(λ)(∗).

Remark 2.2.5. As an immediate consequence of Theorem 2.2.2 we obtain the pseu-dolocality property of the operators:

Let A(λ) ∈ Lµ;ℓ(Rn; Rq;E, E) be given by A(λ) = opx(a)(λ) with a double-

symbol a(x, x′, ξ, λ) ∈ Sµ;ℓ(Rn × Rn × Rn × Rq;E, E), such that a(x, x′, ξ, λ) ≡ 0

for |x− x′| < ε for a sufficiently small ε > 0. Then A(λ) ∈ L−∞(Rn; Rq;E, E).

In particular, if A(λ) ∈ Lµ;ℓ(Rn; Rq;E, E), and ϕ, ψ ∈ C∞b (Rn) such that

dist(suppϕ, suppψ) > 0, then ϕA(λ)ψ ∈ L−∞(Rn; Rq;E, E).

Definition 2.2.6. Let A(λ) = opx(a)(λ) ∈ Lµ;ℓcl (Rn; Rq;E, E), where a ∈

Sµ;ℓcl (Rn×Rn×Rq;E, E). By Theorem 2.2.2 the symbol a is uniquely determined

by A(λ), and so are the homogeneous components of a by (2.1.6). We define

σµ;ℓ∧ (A)(x, ξ, λ) := a(µ)(x, ξ, λ) as the homogeneous component of highest order

and call σµ;ℓ∧ (A) the parameter-dependent homogeneous principal symbol of A(λ)

or simply principal symbol. The mapping A(λ) 7→ σµ;ℓ∧ (A) is continuous.

In case of E = CN− and E = CN+ we write as usual σµ;ℓψ (A) instead of

σµ;ℓ∧ (A).

Remark 2.2.7. With the notations of Theorem 2.2.4 we obtain for classical opera-tors the following relations for the principal symbols of compositions and adjoints:

σµ+µ′;ℓ∧ (AB) = σµ;ℓ

∧ (A)σµ′ ;ℓ

∧ (B) and σµ;ℓ∧ (A(∗)) = σµ;ℓ

∧ (A)∗.This follows from the asymptotic expansions for the Leibniz-product and the

adjoint symbol in Theorem 2.2.4.

Ellipticity and parametrices.

Definition 2.2.8. A symbol a ∈ Sµ;ℓ(Rn × Rn × Rq;E, E) is called parameter-

dependent elliptic, if there is a symbol b ∈ S−µ;ℓ(Rn × Rn × Rq; E, E) such that

a·b− 1 ∈ S−ε;ℓ(Rn × Rn × Rq; E, E),

b·a− 1 ∈ S−ε;ℓ(Rn × Rn × Rq;E,E)

for some ε > 0.Let K ⋐ Rn be compact. A symbol a ∈ Sµ;ℓ(Rn × Rn × Rq;E, E) is called

parameter-dependent elliptic on K, if there is a symbol b ∈ S−µ;ℓ(Rn × Rn ×Rq; E, E) such that ab− 1 and ba− 1 coincide with symbols of order −ε for someε > 0 in a neighbourhood U(K) of K.

In particular we see, that the condition of parameter-dependent ellipticity isnot affected by perturbations of lower-order terms.

An operator A(λ) = opx(a)(λ) ∈ Lµ;ℓ(Rn; Rq;E, E) is called parameter-dependent elliptic (on K), if a is parameter-dependent elliptic (on K).

Remark 2.2.9. The following characterizations of parameter-dependent ellipticityare valid:


a) Let a ∈ Sµ;ℓ(Rn × Rn × Rq;E, E). Then a is parameter-dependent elliptic

if and only if for some R > 0 there exists (a(x, ξ, λ))−1 ∈ L(E, E) for allx ∈ Rn, (ξ, λ) ∈ Rn × Rq with |ξ, λ|ℓ ≥ R, and

sup‖(a(x, ξ, λ))−1‖ 〈ξ, λ〉µℓ ; x ∈ Rn, |ξ, λ|ℓ ≥ R <∞.

If a ∈ Sµ;ℓcl (Rn × Rn × Rq;E, E), then a is parameter-dependent elliptic if and

only if the homogeneous component a(µ)(x, ξ, λ) ∈ L(E, E) of highest order isinvertible for all x ∈ Rn and 0 6= (ξ, λ) ∈ Rn × Rq and

sup‖(a(µ)(x, ξ, λ))−1‖; x ∈ Rn, |ξ, λ|ℓ = 1 <∞.

b) Let a ∈ Sµ;ℓ(Rn × Rn × Rq;E, E) and K ⋐ Rn be compact. Then a isparameter-dependent elliptic on K if and only if for some R > 0 there ex-ists (a(x, ξ, λ))−1 ∈ L(E, E) for all x ∈ K, (ξ, λ) ∈ Rn × Rq with |ξ, λ|ℓ ≥ R,and

sup‖(a(x, ξ, λ))−1‖ 〈ξ, λ〉µℓ ; x ∈ K, |ξ, λ|ℓ ≥ R <∞.

If a ∈ Sµ;ℓcl (Rn × Rn × Rq;E, E), then a is parameter-dependent elliptic on K

if and only if a(µ)(x, ξ, λ) ∈ L(E, E) is invertible for all x ∈ K and 0 6= (ξ, λ) ∈Rn × Rq.

c) Let a ∈ Sµ;ℓcl (Rn × Rn × Rq;E, E). Then a is parameter-dependent elliptic if

and only if there exists b ∈ S−µ;ℓcl (Rn × Rn × Rq; E, E) such that ab − 1 ∈

S−1;ℓcl (Rn × Rn × Rq; E, E) and ba− 1 ∈ S−1;ℓ

cl (Rn × Rn × Rq;E,E).a is parameter-dependent elliptic on a compact set K ⋐ Rn if and only if

there exists b ∈ S−µ;ℓcl (Rn×Rn×Rq; E, E) such that ab− 1 and ba− 1 coincide

with classical symbols of order −1 in a neighbourhood U(K) of K.

Proof. Note first that in view of Definition 2.2.8 the conditions in a) and b) areclearly necessary for parameter-dependent ellipticity. To prove the sufficiency ofthe conditions in a) let χ ∈ C∞(Rn × Rq) such that χ ≡ 0 for |ξ, λ|ℓ ≤ R+ 1 andχ ≡ 1 for |ξ, λ|ℓ ≥ R+ 2. For (x, ξ, λ) ∈ Rn × Rn × Rq define

b(x, ξ, λ) :=

χ(ξ, λ)(a(x, ξ, λ))−1 in the general case

χ(ξ, λ)(a(µ)(x, ξ, λ))−1 in the classical case.

Thus we see that b ∈ S−µ;ℓ(cl) (Rn×Rn×Rq; E, E), and moreover ab−1 ∈ S−1;ℓ

(cl) (Rn×Rn × Rq; E, E) and ba − 1 ∈ S−1;ℓ

(cl) (Rn × Rn × Rq;E,E). This proves a) and the

first assertion in c).Now assume that the conditions in b) hold. Note that they not only hold for

x ∈ K, but also in a neighbourhood V (K) of K. Let ϕ ∈ C∞0 (V (K)) with ϕ ≡ 1 in

a neighbourhood U(K). Let χ ∈ C∞(Rn ×Rq) such that χ ≡ 0 for |ξ, λ|ℓ ≤ R+ 1and χ ≡ 1 for |ξ, λ|ℓ ≥ R+ 2. For (x, ξ, λ) ∈ Rn × Rn × Rq define

b(x, ξ, λ) :=

χ(ξ, λ)ϕ(x)(a(x, ξ, λ))−1 in the general case

χ(ξ, λ)ϕ(x)(a(µ)(x, ξ, λ))−1 in the classical case.


We thus see that b ∈ S−µ;ℓ(cl) (Rn ×Rn × Rq; E, E) with ab− ϕI ∈ S−1;ℓ

(cl) (Rn × Rn ×Rq; E, E) and ba− ϕI ∈ S−1;ℓ

(cl) (Rn×Rn×Rq;E,E) which shows b) and completes

the proof of c).

Theorem 2.2.10. Let A(λ) ∈ Lµ;ℓ(Rn; Rq;E, E). Then the following are equiva-lent:

a) A(λ) is parameter-dependent elliptic.

b) There exists an operator P (λ) ∈ L−µ;ℓ(Rn; Rq; E, E), such that A(λ)P (λ)−1 ∈L−ε;ℓ(Rn; Rq; E, E) and P (λ)A(λ) − 1 ∈ L−ε;ℓ(Rn; Rq;E,E) for some ε > 0.

c) There exists an operator P (λ) ∈ L−µ;ℓ(Rn; Rq; E, E), such that A(λ)P (λ)−1 ∈L−∞(Rn; Rq; E, E) and P (λ)A(λ) − 1 ∈ L−∞(Rn; Rq;E,E).

If even A(λ) ∈ Lµ;ℓcl (Rn; Rq;E, E) is parameter-dependent elliptic then every P (λ)

satisfying c) belongs to L−µ;ℓcl (Rn; Rq; E, E).

Every P (λ) ∈ L−µ;ℓ(cl) (Rn; Rq; E, E) satisfying c) is called a (parameter-

dependent) parametrix of A(λ).

Proof. Assume that a) holds. Let A(λ) = opx(a)(λ) with a ∈ Sµ;ℓ(Rn × Rn ×Rq;E, E). Let b ∈ S−µ;ℓ(Rn×Rn×Rq; E, E) satisfying the condition of Definition2.2.8. Now the asymptotic expansion of the Leibniz-product in Theorem 2.2.4(2.2.2) gives that b#a− 1 ∈ S−ε;ℓ(Rn×Rn×Rq;E,E) and a#b− 1 ∈ S−ε;ℓ(Rn×Rn×Rq; E, E) for some ε > 0 which implies b). If a ∈ Sµ;ℓ

cl (Rn×Rn×Rq;E, E) we

choose b ∈ S−µ;ℓcl (Rn×Rn×Rq; E, E) satisfying condition c) of Remark 2.2.9. We

then even obtain b#a− 1 ∈ S−1;ℓcl (Rn×Rn×Rq;E,E) and a#b− 1 ∈ S−1;ℓ

cl (Rn×Rn × Rq; E, E).

Now assume that b) is fulfilled. Let P (λ) = opx(b)(λ) and A(λ)P (λ) =

1 − opx(r)(λ) with r ∈ S−ε;ℓ(Rn × Rn × Rq; E, E). From Theorem 2.1.8 and

Theorem 2.2.4 we see that there is a symbol c ∈ S−ε;ℓ(Rn × Rn × Rq; E, E)such that c ∼ ∑

j∈N

#(j)r. Now define P(R)(λ) := opx(b#(1 + c))(λ). Then we

have A(λ)P(R)(λ) − 1 ∈ L−∞(Rn; Rq; E, E) as desired. Analogously, we obtaina parametrix P(L)(λ) from the left. But both the left- and the right-parametrix

differ only by a term in L−∞(Rn; Rq; E, E) which we see from considering theproduct P(L)(λ)A(λ)P(R)(λ). This implies c). Note that if we had started with thecase ε = 1 and P (λ) as well as the remainder being classical, we would have ob-tained also a classical parametrix which proves the second assertion of the theorem(cf. Remark 2.1.10).

c) implies a) follows at once from Theorem 2.2.4.

Corollary 2.2.11. Let A(λ) ∈ Lµ;ℓ(cl)(R

n; Rq;E, E) and K ⋐ Rnx be compact.

Then A(λ) is parameter-dependent elliptic on K if and only if there are ϕ, ψ ∈C∞

0 (Rn) such that ϕψ = ϕ, ϕ ≡ 1 on K, and P (λ) ∈ L−µ;ℓ(cl) (Rn; Rq; E, E)


such that ϕ(A(λ)P (λ) − 1)ψ ∈ L−∞(Rn; Rq; E, E) and ϕ(P (λ)A(λ) − 1)ψ ∈L−∞(Rn; Rq;E,E).

Sobolev spaces and continuity.

Definition 2.2.12. Let E be a Hilbert space. For s ∈ R the Sobolev spaceHs(Rn, E) is defined to consist of all u ∈ S′(Rn, E) such that Fu is a regulardistribution, and

‖u‖Hs(Rn,E) :=(∫

Rn

〈ξ〉2s‖Fu(ξ)‖2E dξ

) 12

<∞.

In case of E = C the space is suppressed from the notation.

Theorem 2.2.13. Let E and E be Hilbert spaces. Let a ∈ Sµ;ℓ(Rn×Rn×Rq;E, E)and s, ν ∈ R where ν ≥ µ. Then opx(a)(λ) extends for λ ∈ Rq by continuity to

an operator opx(a)(λ) ∈ L(Hs(Rn, E), Hs−ν(Rn, E)), and we have the followingestimate for the norm:

‖opx(a)(λ)‖L(Hs(Rn,E),Hs−ν(Rn,E)) ≤Cs,ν〈λ〉

µℓ ν ≥ 0

Cs,ν〈λ〉µ−νℓ ν ≤ 0,

(2.2.5)

where Cs,ν > 0 is a constant depending on s, ν and a.More precisely, this induces a continuous embedding

Lµ;ℓ(Rn; Rq;E, E) →Sµℓ (Rq;Hs(Rn, E), Hs−ν(Rn, E)) ν ≥ 0

Sµ−νℓ (Rq;Hs(Rn, E), Hs−ν(Rn, E)) ν ≤ 0

(2.2.6)

into the space of operator-valued symbols in the Sobolev spaces.

Coordinate invariance.

Definition 2.2.14. Let U ⊆ Rn be an open set. Then A(λ) ∈ Lµ;ℓ(Rn; Rq;E, E)is said to be compactly supported in U, if for some ϕ, ψ ∈ C∞

0 (U), and some

B(λ) ∈ Lµ;ℓ(Rn; Rq;E, E), we have A(λ) = ϕB(λ)ψ.In other words: A(λ) is compactly supported in U if and only if there is a

compact set K ⊆ U×U such that

suppKA(λ) ⊆ K for all λ ∈ Rq, (2.2.7)

where KA(λ) ∈ S′(Rn×Rn,L(E, E)) denotes the operator-valued Schwartz kernelof the operator A(λ).

For each compact set K ⊆ U×U the space of compactly supported op-

erators A(λ) ∈ Lµ;ℓ(cl)(R

n; Rq;E, E) satisfying (2.2.7) is a closed subspace of

Lµ;ℓ(cl)(R

n; Rq;E, E).

Let Lµ;ℓcomp (cl)(U ; Rq;E, E) denote the space of all (classical) parameter-

dependent pseudodifferential operators that are compactly supported in U . Weendow this space with the inductive limit topology of the subspaces of operators


with Schwartz kernels satisfying (2.2.7) (taken over all compact sets K ⊆ U×U).Thus it becomes a strict countable inductive limit of Frechet spaces.

Note that A(λ) = opx(a)(λ) ∈ Lµ;ℓcomp(U ; Rq;E, E) acts as a family of contin-

uous operators A(λ) : C∞0 (U,E) → C∞

0 (U, E), and its symbol a(x, ξ, λ) is uniquelydetermined by this action.

Theorem 2.2.15. Let U, V ⊆ Rn be open subsets and χ : U → V a diffeomor-phism. Then the operator pull-back χ∗A(λ) defined as

(χ∗A(λ))u := χ∗(A(λ)(χ∗u)) (2.2.8)

for u ∈ C∞0 (U,E) and A(λ) ∈ Lµ;ℓ

comp(V ; Rq;E, E), with the pull-back χ∗

and push-forward χ∗ for C∞0 -functions, defines a topological isomorphism χ∗ :

Lµ;ℓcomp (cl)(V ; Rq;E, E) → Lµ;ℓ

comp (cl)(U ; Rq;E, E).

Moreover, given A(λ) = opx(a)(λ) ∈ Lµ;ℓcomp (cl)(V ; Rq;E, E), then χ∗A(λ) =

opx(b)(λ) with a symbol b ∈ Sµ;ℓ(cl)(R

n×Rn×Rq;E, E) having the following asymp-

totic expansion in terms of a and χ:

b(x, ξ, λ) ∼∑

α∈Nn0

(∂αξ a)(χ(x), [Dχ(x)−1 ]tξ, λ)ϕα(x, ξ) (2.2.9)

with polynomials ϕα(x, ξ) in ξ of degree less or equal to |α|2 and ϕ0 ≡ 1, that are

given completely in terms of the diffeomorphism χ.Note that the symbol a vanishes identically outside a compact set in V which

gives a meaning to this asymptotic expansion.In particular, we obtain b(x, ξ, λ) − a(χ(x), [Dχ(x)−1 ]tξ, λ) ∈ Sµ−1;ℓ(Rn ×

Rn × Rq;E, E). This yields in the classical case to the following relation for theprincipal symbols:

σµ;ℓ∧ (χ∗A)(x, ξ, λ) = σµ;ℓ

∧ (A)(χ(x), [Dχ(x)−1 ]tξ, λ). (2.2.10)

This also shows, that χ∗A(λ) is parameter-dependent elliptic on a compact setK ⊆ U if and only if A(λ) is parameter-dependent elliptic on χ(K) ⊆ V .

2.3. Parameter-dependent Volterra symbols

Let

H := z ∈ C; Im(z) ≥ 0 ⊆ C ∼= R2

be the upper half-plane in C. The significant property of Volterra operators, resp.symbols with the Volterra property is, that in addition to the symbol estimateswe employ the analyticity in the interior of H.

Definition 2.3.1. Let E and E be Hilbert spaces. For µ ∈ R we define

Sµ;ℓV (cl)(R

n × H;E, E) := Sµ;ℓ(cl)(R

n × H;E, E) ∩A(

H, C∞(Rn,L(E, E))),


which is a closed subspace of Sµ;ℓ(cl)(R

n × H;E, E). Analogously, we define

S−∞V (Rn × H;E, E) :=

⋂

µ∈R

Sµ;ℓV (Rn × H;E, E),

as well as the spaces of x- (resp. x′-) and (x, x′)-dependent symbols

Sµ;ℓV (cl)(R

n × Rn × H;E, E) := C∞b (Rn, Sµ;ℓ

V (cl)(Rn × H;E, E)),

Sµ;ℓV (cl)(R

n × Rn × Rn × H;E, E) := C∞b (Rn × Rn, Sµ;ℓ

V (cl)(Rn × H;E, E)).

These symbols are called symbols with the Volterra property or simply Volterrasymbols which is indicated by the subscript V .

Of course, this notion also applies to the case of scales of Hilbert spacesinvolved instead of the single spaces only, and we shall employ the same conventionsas in the case without the extra analyticity condition, see Definition 2.1.3.

Proposition 2.3.2. a) The restriction of the parameter to the real line induces

a continuous embedding Sµ;ℓV (cl)(R

n × H;E, E) → Sµ;ℓ(cl)(R

n × R;E, E).

b) The homogeneous components of a symbol a ∈ Sµ;ℓV cl(R

n×H;E, E) are analytic

in

H.

Kernel cut-off and asymptotic expansion of Volterra symbols.

Definition 2.3.3. Let Ej and Ej be scales of Hilbert spaces. For short, weset

E := ind-limj∈N


Ej .

Let (µk) ⊆ R be a sequence of reals such that µk −→k→∞

−∞ and µ := maxk∈N

µk.

Moreover, let ak ∈ Sµk;ℓV (Rn×Rn×H; E , E). A symbol a ∈ Sµ;ℓV (Rn×Rn×H; E , E)

is called the asymptotic expansion of the ak, if for every R ∈ R there is a k0 ∈ N

such that for k > k0

a−k∑

j=1

aj ∈ SR;ℓV (Rn × Rn × H; E , E).

The symbol a is uniquely determined modulo S−∞V (Rn × Rn × H; E , E).

For short we write a ∼V

∞∑j=1

aj .

Remark 2.3.4. Note that the notion of asymptotic expansion for Volterra symbolsfrom Definition 2.3.3 is more refined than that of Definition 2.1.6. What makesthings more complicated is the extra analyticity condition. In particular, the stan-dard excision function arguments in the proof of the existence of symbols having aprescribed asymptotic expansion, see also Lemma 2.1.7, cannot be applied to ob-tain corresponding existence results in the Volterra case. The substitute for theseare kernel cut-off techniques, see also Proposition 2.3.8.


Definition 2.3.5. Let E and E be Hilbert spaces, and let ϕ ∈ C∞b (R). On

Sµ;ℓ(Rn×R;E, E) define the kernel cut-off operator H(ϕ) by means of the follow-ing oscillatory integral:

(H(ϕ)a

)(ξ, λ) :=

∫

R

∫

R

e−itτϕ(t)a(ξ, λ − τ) dt dτ (2.3.1)

for a ∈ Sµ;ℓ(Rn × R;E, E).

Theorem 2.3.6. Let Ej and Ej be scales of Hilbert spaces. We again use

the abbreviations E for the inductive limit of the spaces Ej, as well as E for the

projective limit of the spaces Ej .Then the mapping

H :

C∞b (R)×Sµ;ℓ

(cl)(Rn × R; E , E) −→ Sµ;ℓ

(cl)(Rn × R; E , E),

C∞b (R)×Sµ;ℓ

V (cl)(Rn × H; E , E) −→ Sµ;ℓ

V (cl)(Rn × H; E , E)

is bilinear and continuous.The following asymptotic expansion holds for H(ϕ)a in terms of ϕ and a:

H(ϕ)a ∼(V )

∞∑

k=0

((−1)k

k!Dkt ϕ(0)

)· ∂kλa (2.3.2)

where ∂λ denotes the complex derivative with respect to λ ∈ H in case of Volterrasymbols.

Corollary 2.3.7. Let ϕ ∈ C∞0 (R) with ϕ ≡ 1 near t = 0. Then the operator

I −H(ϕ) is coninuous in the spaces

Sµ;ℓ(Rn × R; E , E) → S−∞(Rn × R; E , E),

Sµ;ℓV (Rn × H; E , E) → S−∞

V (Rn × H; E , E).

Proposition 2.3.8. Let (µk) ⊆ R such that µk ≥ µk+1 −→k→∞

−∞. Furthermore,

for each k ∈ N let (Akj )j∈N ⊆ Sµk;ℓV (Rn×H; E , E) be a countable system of boundedsets. Let ϕ ∈ C∞

0 (R), and for c ∈ [1,∞) let ϕc ∈ C∞0 (R) be defined as ϕc(t) :=

ϕ(ct).Then there is a sequence (ci) ⊆ [1,∞) with ci < ci+1 −→

i→∞∞ such that for

each k ∈ N∞∑

i=k

supa∈Aij

p(H(ϕdi)a

)<∞ (2.3.3)

for all continuous seminorms p on Sµk;ℓV (Rn × H; E , E) and every j ∈ N, and forall sequences (di) ⊆ R+ with di ≥ ci.

Theorem 2.3.9. Let Ej and Ej be scales of Hilbert spaces, and E and Eas before. Let (µk) ⊆ R such that µk −→


k∈N

µk. Moreover, let

ak ∈ Sµk;ℓV (Rn×Rn×H; E , E). Then there exists a ∈ Sµ;ℓV (Rn×Rn×H; E , E) such


that a ∼V

∞∑j=1

aj. The asymptotic sum a is uniquely determined modulo S−∞V (Rn ×

Rn × H; E , E).

If the sequence (µk)k∈N0 is given as µk = µ− k and ak ∈ Sµ−k;ℓV cl (Rn × Rn ×H; E , E), then also a ∈ Sµ;ℓ

V cl(Rn × Rn × H; E , E).

Proof. For the proof we may without loss of generality assume that µk ≥ µk+1 −→k→∞

−∞. For k, j ∈ N let

Akj :=∂αx ak(x); x ∈ Rn, |α| ≤ j

.

Then Akj ⊆ Sµk;ℓV (Rn × H; E , E) is bounded. Let ϕ ∈ C∞0 (R) such that ϕ ≡ 1

near t = 0. Now apply Proposition 2.3.8. With a suitable sequence (ci) ⊆ [1,∞)formula (2.3.3) becomes

∞∑

i=k

supp(H(ϕci)(∂

αx ai(x))

); x ∈ Rn, |α| ≤ j

<∞

for all continuous seminorms p on Sµk;ℓV (Rn × H; E , E).

This shows that∞∑i=k

H(ϕci)ai is unconditionally convergent in Sµk;ℓV (Rn ×

Rn × H; E , E) for every k ∈ N. Now define

a :=

∞∑

i=1

H(ϕci)ai ∈ Sµ1;ℓV (Rn × Rn × H; E , E).

We thus see

a−k∑

i=1

ai =

∞∑

i=k+1

H(ϕci)ai −k∑

i=1

(I −H(ϕci)

)ai

wherek∑

i=1

(I −H(ϕci)

)ai ∈ S−∞

V (Rn × Rn × H; E , E)

in view of Corollary 2.3.7. This yields the desired result, since the uniquenessassertion is clear.

The translation operator in Volterra symbols.

Definition 2.3.10. For z = iτ ∈ iR ⊆ C, τ ≥ 0, define the translation operator

Tiτ on Sµ;ℓV (Rn × H; E , E) via

(Tiτa

)(ξ, λ) := a(ξ, λ+ iτ).

Proposition 2.3.11. For every τ ≥ 0 the translation operator Tiτ acts linear andcontinuous in the spaces

Tiτ : Sµ;ℓV (cl)(R

n × H; E , E) −→ Sµ;ℓV (cl)(R

n × H; E , E).


Moreover, Tiτa has the following asymptotic expansion in terms of τ and a:

Tiτa ∼V

∞∑

k=0

(iτ)k

k!· ∂kλa.

In particular, the operator I − Tiτ is continuous in the spaces

I − Tiτ : Sµ;ℓV (cl)(R

n × H; E , E) −→ Sµ−ℓ;ℓV (cl) (Rn × H; E , E).

Notation 2.3.12. For µ ∈ R let S(µ;ℓ)((

Rn × H)\ 0;E, E

)denote the closed

subspace of C∞((Rn×H

)\0,L(E, E)) consisting of all anisotropic homogeneous

functions of degree µ. Moreover, let

S(µ;ℓ)V

((Rn × H

)\ 0;E, E

):= S(µ;ℓ)

((Rn × H

)\ 0;E, E

)

∩A(

H, C∞(Rn,L(E, E))),

which is a closed subspace of S(µ;ℓ)((

Rn × H)\ 0;E, E

).

Theorem 2.3.13. For every τ > 0 the mapping Tiτ : a(ξ, λ) 7→ a(ξ, λ + iτ) iscontinuous in the spaces

Tiτ : S(µ;ℓ)V

((Rn × H

)\ 0;E, E

)−→ Sµ;ℓ

V cl(Rn × H;E, E).

Moreover, for every 0-excision function χ ∈ C∞(Rn×H), the following asymptoticexpansion holds for Tiτa:

Tiτa ∼∞∑

k=0

(iτ)k

k!· χ

(∂kλa

).

This shows, in particular, that for the homogeneous component of order µ we havethe identity

(Tiτa

)(µ)

= a.

In other words, the restriction of the “principal symbol sequence” (on symboliclevel) to Volterra symbols is topologically exact and splits:

0 −→Sµ−1;ℓV cl (Rn × H;E, E) −→ Sµ;ℓ

V cl(Rn × H;E, E) −→

S(µ;ℓ)V

((Rn × H

)\ 0;E, E

)−→ 0.

The operator Tiτ provides a splitting of this sequence. Analogous assertions holdin case of scales of Hilbert spaces involved.

2.4. Parameter-dependent Volterra operators

Remark 2.4.1. In this section we first recall the basic elements of pseudodifferentialcalculus on Rn built upon parameter-dependent Volterra symbols, which is ratherstraightforward in view of Sections 2.2 and 2.3.

Secondly, we study parabolicity of such Volterra operators which is defined byrequiring the parameter-dependent ellipticity of the symbols. The main point hereis that we are in need to construct a parametrix which itself has again the Volterraproperty. The latter cannot be obtained from Theorem 2.2.10 (or its proof) because


there are arguments with excision functions involved, which destroy the analyticityin the interior of the half-plane (see also Remark 2.2.9). However, the possibility tocarry out asymptotic expansions, see Theorem 2.3.9, which relies on kernel cut-offtechniques, and the translation operator in Volterra symbol spaces provide thetools to handle these difficulties.

Definition 2.4.2. Let E and E be Hilbert spaces. For µ ∈ R the space of Volterrapseudodifferential operators respectively operators with the Volterra property isdefined as

Lµ;ℓV (cl)(R

n; H;E, E) := opx(a)(λ); a ∈ Sµ;ℓV (cl)(R

n × Rn × Rn × H;E, E)⊆ Lµ;ℓ

(cl)(Rn; H;E, E).

In case of E = E = C the spaces are suppressed from the notation as usual.


Theorem 2.4.3. Let a ∈ Sµ;ℓV (cl)(R

n × Rn × Rn × H;E, E). Then the unique

left- and right-symbol from Theorem 2.2.2 associated to the operator opx(a)(λ) ∈Lµ;ℓV (cl)(R

n; H;E, E) are Volterra symbols, i. e., we have aL(x, ξ, λ), aR(x′, ξ, λ) ∈Sµ;ℓV (cl)(R

n × Rn × H;E, E).

Moreover, the asymptotic expansions for aL and aR in terms of a are validin the Volterra sense:

aL(x, ξ, λ) ∼V

∑

α∈Nn0

1

α!∂αξ D

αx′a(x, x′, ξ, λ)|x′=x,

aR(x′, ξ, λ) ∼V

∑

α∈Nn0

1

α!(−1)|α|∂αξ D

αxa(x, x

′, ξ, λ)|x=x′ .

Remark 2.4.4. By Theorem 2.4.3 the mapping opx restricts to an isomorphismbetween the space of Volterra left-symbols and Volterra pseudodifferential opera-tors:

Sµ;ℓV (cl)(R

n × Rn × H;E, E)opx−→∼=

Lµ;ℓV (cl)(R

n; H;E, E).

Consequently, Lµ;ℓV (cl)(R

n; H;E, E) is a closed subspace of Lµ;ℓ(cl)(R

n; H;E, E).

The space of parameter-dependent Volterra operators of order −∞ is denotedby

L−∞V (Rn; H;E, E) =

⋂

µ∈R

Lµ;ℓV (Rn; H;E, E)

= opx(a)(λ); a ∈ S−∞V (Rn × Rn × H;E, E).

In view of Proposition 2.3.2 the restriction of the parameter to the real lineinduces a continuous embedding

Lµ;ℓV (cl)(R

n × H;E, E) → Lµ;ℓ(cl)(R

n × R;E, E).


Theorem 2.4.5. Let E, E and E be Hilbert spaces, and A(λ) = opx(a)(λ) ∈Lµ;ℓV (cl)(R

n; H; E, E) as well as B(λ) = opx(b)(λ) ∈ Lµ′;ℓV (cl)(R

n; H;E, E) with a ∈Sµ;ℓV (cl)(R

n × Rn × H; E, E) and b ∈ Sµ′;ℓ

V (cl)(Rn × Rn × H;E, E).

Then the composition as operators on S(Rn, E) belongs to the operator-space

Lµ+µ′;ℓV (cl) (Rn; H;E, E), i. e., the Leibniz-product a#b of the symbols a and b (cf.

Theorem 2.2.4) belongs to Sµ+µ′;ℓV (cl) (Rn × Rn × H;E, E).

Moreover, for a#b the following asymptotic expansion holds:

a#b ∼V

∑

α∈Nn0

1

α!(∂αξ a)(D

αx b). (2.4.1)

Remark 2.4.6. From Theorem 2.4.3 we conclude the pseudolocality property ofthe parameter-dependent Volterra calculus (see also Remark 2.2.5):

Let A(λ) ∈ Lµ;ℓV (Rn; H;E, E) be given by A(λ) = opx(a)(λ) with a double-

symbol a(x, x′, ξ, λ) ∈ Sµ;ℓV (Rn × Rn × Rn × H;E, E), such that a(x, x′, ξ, λ) ≡ 0

for |x− x′| < ε for a sufficiently small ε > 0. Then A(λ) ∈ L−∞V (Rn; H;E, E).

In particular, if ϕ, ψ ∈ C∞b (Rn) such that dist(suppϕ, suppψ) > 0, then

ϕA(λ)ψ ∈ L−∞V (Rn; H;E, E). The mapping Lµ;ℓ

V (Rn; H;E, E) ∋ A(λ) 7−→ϕA(λ)ψ ∈ L−∞

V (Rn; H;E, E) is continuous.

Theorem 2.4.7. For every µ ∈ R the principal symbol sequence in Volterra pseu-dodifferential operators is topologically exact and splits:

0 −→Lµ−1;ℓV cl (Rn; H;E, E)

ı−→ Lµ;ℓV cl(R

n; H;E, E)σµ;ℓ∧−→

C∞b

(Rn, S

(µ;ℓ)V

((Rn × H

)\ 0;E, E

))−→ 0.

The translation operator Tiτ for τ > 0 gives rise to a splitting of this sequence.Analogous assertions hold in case of scales of Hilbert spaces involved.

Continuity and coordinate invariance.

Theorem 2.4.8. Let E and E be Hilbert spaces, and let a ∈ Sµ;ℓV (Rn × Rn ×

H;E, E), as well as s, ν ∈ R with ν ≥ µ. Then opx(a)(λ) extends for λ ∈ H by

continuity to an operator opx(a)(λ) ∈ L(Hs(Rn, E), Hs−ν(Rn, E)), which inducesa continuous embedding

Lµ;ℓV (Rn; H;E, E) →

Sµℓ

V (H;Hs(Rn, E), Hs−ν(Rn, E)) ν ≥ 0

Sµ−νℓ

V (H;Hs(Rn, E), Hs−ν(Rn, E)) ν ≤ 0(2.4.2)

into the space of operator-valued Volterra symbols in the Sobolev spaces.

Remark 2.4.9. Let U ⊆ Rn be an open set. Recall from Section 2.2 that an operatorA(λ) ∈ Lµ;ℓ(Rn; H;E, E) is compactly supported in U if and only if there is acompact set K ⊆ U×U such that

suppKA(λ) ⊆ K for all λ ∈ H (2.4.3)


where KA(λ) ∈ S′(Rn×Rn,L(E, E)) denotes the operator-valued Schwartz kernelof the operator A(λ).

For each compact set K ⊆ U×U the space of compactly supported Volterra

operators A(λ) ∈ Lµ;ℓV (cl)(R

n; H;E, E) satisfying (2.4.3) is a closed subspace of

Lµ;ℓV (cl)(R

n; H;E, E).

Let Lµ;ℓcompV (cl)(U ; H;E, E) denote the space of all (classical) parameter-

dependent Volterra pseudodifferential operators that are compactly supported inU . This space is endowed with the inductive limit topology of the subspaces ofoperators with Schwartz kernels satisfying (2.4.3) (taken over all compact sets

K ⊆ U×U), and thus it is a closed subspace of Lµ;ℓcomp (cl)(U ; H;E, E).

Theorem 2.4.10. Let U, V ⊆ Rn be open subsets and χ : U → V a diffeomor-phism. In view of Theorem 2.2.15 the operator pull-back χ∗ (cf. (2.2.8)) inducesa topological isomorphism

χ∗ : Lµ;ℓcomp (cl)(V ; H;E, E) → Lµ;ℓ

comp (cl)(U ; H;E, E).

Its restriction to the spaces of compactly supported Volterra pseudodifferential op-erators acts as a topological isomorphism

χ∗ : Lµ;ℓcompV (cl)(V ; H;E, E) → Lµ;ℓ

compV (cl)(U ; H;E, E).

Moreover, given A(λ) = opx(a)(λ) ∈ Lµ;ℓcompV (cl)(V ; H;E, E), then χ∗A(λ) =

opx(b)(λ) with a symbol b ∈ Sµ;ℓV (cl)(R

n × Rn × H;E, E) having the following as-

ymptotic expansion in the Volterra sense in terms of a and χ:

b(x, ξ, λ) ∼V

∑

α∈Nn0

(∂αξ a)(χ(x), [Dχ(x)−1]tξ, λ)ϕα(x, ξ) (2.4.4)

with the universal polynomials ϕα(x, ξ) in ξ of degree less or equal to |α|2 and

ϕ0 ≡ 1 depending only on the diffeomorphism χ from the asymptotic expansion(2.2.9) in Theorem 2.2.15.

In particular, we obtain b(x, ξ, λ) − a(χ(x), [Dχ(x)−1 ]tξ, λ) ∈ Sµ−1;ℓV (Rn ×

Rn × H;E, E).

Parabolicity for Volterra pseudodifferential operators.

Definition 2.4.11. A symbol a ∈ Sµ;ℓV (Rn × Rn × H;E, E) is called parabolic, if

a is parameter-dependent elliptic as an element in Sµ;ℓ(Rn × Rn × H;E, E).

Let K ⋐ Rn be compact. A symbol a ∈ Sµ;ℓV (Rn × Rn × H;E, E) is called

parabolic on K, if a is parameter-dependent elliptic on K as an element inSµ;ℓ(Rn × Rn × H;E, E).

An operator A(λ) = opx(a)(λ) ∈ Lµ;ℓV (Rn; H;E, E) is called parabolic (on

K), if a is parabolic (on K).

Proposition 2.4.12. Let a ∈ Sµ;ℓV (cl)(R

n × Rn × H;E, E).


a) a is parabolic if and only if there exists an element b ∈ S−µ;ℓV (cl)(R

n×Rn×H; E, E)

such that

a·b− 1 ∈ S−1;ℓV (cl)(R

n × Rn × H; E, E),

b·a− 1 ∈ S−1;ℓV (cl)(R

n × Rn × H;E,E).

b) a is parabolic on a compact set K ⋐ Rn if and only if there exists an element

b ∈ S−µ;ℓV (cl)(R

n × Rn × H; E, E) such that a·b − 1 and b·a − 1 coincide with

(classical) Volterra symbols of order −1 in a neighbourhood U(K) of K.

Proof. We only have to prove the necessity of the conditions in a) and b),for the sufficiency follows immediately from the definition of parabolicity asparameter-dependent ellipticity (see Definition 2.2.8 and Remark 2.2.9). Assume

that a ∈ Sµ;ℓV (cl)(R

n × Rn × H;E, E) is parabolic. According to Remark 2.2.9,

for some sufficiently large R > 0 there exists (a(x, ξ, λ))−1 ∈ L(E, E) for allx ∈ Rn, (ξ, λ) ∈ Rn × H with |ξ, λ|ℓ ≥ R, and

sup‖(a(x, ξ, λ))−1‖ 〈ξ, λ〉µℓ ; x ∈ Rn, |ξ, λ|ℓ ≥ R <∞.

Consequently, if we choose τ ∈ R+ sufficiently large, we conclude that for all

x ∈ Rn and all (ξ, λ) ∈ Rn × H there exists((Tiτa)(x, ξ, λ)

)−1 ∈ L(E, E) with

sup‖((Tiτa)(x, ξ, λ)

)−1‖ 〈ξ, λ〉µℓ ; x ∈ Rn, (ξ, λ) ∈ Rn × H <∞

for the symbol Tiτa ∈ Sµ;ℓV (cl)(R

n × Rn × H;E, E) (cf. Proposition 2.3.11). Recall

that a− Tiτa ∈ Sµ−ℓ;ℓV (cl) (Rn × Rn × H;E, E). Consequently we see, using Theorem

2.3.13, that the function

b(x, ξ, λ) :=

((Tiτa)(x, ξ, λ)

)−1in the general case

Tiτ(a(µ)

)−1(x, ξ, λ) in the classical case

belongs to S−µ;ℓV (cl)(R

n × Rn × H; E, E) and satisfies the asserted condition in a).

Now assume that a ∈ Sµ;ℓV (cl)(R

n × Rn × H;E, E) is parabolic on a compact

set K ⋐ Rn. Employing again Remark 2.2.9 we see, that there is a neighbourhoodV (K) of K and a sufficiently large R > 0, such that there exists (a(x, ξ, λ))−1 ∈L(E, E) for all x ∈ V (K), (ξ, λ) ∈ Rn × H with |ξ, λ|ℓ ≥ R, and

sup‖(a(x, ξ, λ))−1‖ 〈ξ, λ〉µℓ ; x ∈ V (K), |ξ, λ|ℓ ≥ R <∞.

Passing as before to the symbol Tiτa for sufficiently large τ ∈ R+ we see, that there

exists((Tiτa)(x, ξ, λ)

)−1 ∈ L(E, E) for all x ∈ V (K) and all (ξ, λ) ∈ Rn ×H, and

sup‖((Tiτa)(x, ξ, λ)

)−1‖ 〈ξ, λ〉µℓ ; x ∈ V (K), (ξ, λ) ∈ Rn × H <∞.

In the classical case we have

sup‖Tiτ(a(µ)

)−1(x, ξ, λ)‖ 〈ξ, λ〉µℓ ; x ∈ V (K), (ξ, λ) ∈ Rn × H <∞.


Now choose a function ϕ ∈ C∞0 (V (K)) such that ϕ ≡ 1 in a neighbourhood U(K)

of K, and define

b(x, ξ, λ) :=

ϕ(x)

((Tiτa)(x, ξ, λ)

)−1in the general case

ϕ(x)Tiτ(a(µ)

)−1(x, ξ, λ) in the classical case.

Then b belongs to S−µ;ℓV (cl)(R

n ×Rn ×H; E, E) and fulfills the asserted condition in

b).

Theorem 2.4.13. Let A(λ) ∈ Lµ;ℓV (Rn; H;E, E). Then the following are equiva-

lent:

a) A(λ) is parabolic.

b) There exists an operator P (λ) ∈ L−µ;ℓV (Rn; H; E, E), such that A(λ)P (λ)− 1 ∈

L−ε;ℓV (Rn; H; E, E) and P (λ)A(λ) − 1 ∈ L−ε;ℓ

V (Rn; H;E,E) for some ε > 0.

c) There exists an operator P (λ) ∈ L−µ;ℓV (Rn; H; E, E), such that A(λ)P (λ)− 1 ∈

L−∞V (Rn; H; E, E) and P (λ)A(λ) − 1 ∈ L−∞

V (Rn; H;E,E).

If A(λ) ∈ Lµ;ℓV cl(R

n; H;E, E) is parabolic, then every P (λ) satisfying c) belongs

to L−µ;ℓV cl (Rn; H; E, E). Every P (λ) ∈ L−µ;ℓ

V (cl)(Rn; H; E, E) satisfying c) is called a

Volterra parametrix of A(λ).

Proof. In view of Definition 2.4.11 of parabolicity for Volterra pseudodifferentialoperators and Theorem 2.2.10 it suffices to show that a) implies b), and b) impliesc).

Assume that a) holds. Let A(λ) = opx(a)(λ) with a ∈ Sµ;ℓV (Rn×Rn×H;E, E).

Let b ∈ S−µ;ℓV (Rn × Rn × H; E, E) satisfying the condition in a) of Proposition

2.4.12. Now the asymptotic expansion (2.4.1) in the Volterra sense of the Leibniz-

product in Theorem 2.4.5 gives that b#a − 1 ∈ S−1;ℓV (Rn × Rn × H;E,E) and

a#b− 1 ∈ S−1;ℓV (Rn × Rn × H; E, E) which implies b).

Now assume that b) is fulfilled. Let P (λ) = opx(b)(λ) and A(λ)P (λ) =

1 − opx(r)(λ) with r ∈ S−ε;ℓV (Rn × Rn × H; E, E). From Theorem 2.3.9 and The-

orem 2.4.5 we see that there is a symbol c ∈ S−ε;ℓV (Rn × Rn × H; E, E) such

that c ∼V

∑j∈N

#(j)r. Now define P(R)(λ) := opx(b#(1 + c))(λ). Then we have

A(λ)P(R)(λ)−1 ∈ L−∞V (Rn; H; E, E) as desired. Analogously, we obtain a Volterra

parametrix P(L)(λ) from the left. But both the left- and the right-parametrix differ

only by a term in L−∞V (Rn; H; E, E) which follows from considering the product

P(L)(λ)A(λ)P(R)(λ). This implies c).

Corollary 2.4.14. Let A(λ) ∈ Lµ;ℓV (cl)(R

n; H;E, E) and K ⋐ Rn be compact. Then

A(λ) is parabolic on K if and only if there are ϕ, ψ ∈ C∞0 (Rn) such that ϕψ = ϕ,

ϕ ≡ 1 on K, and P (λ) ∈ L−µ;ℓV (cl)(R

n; H; E, E) such that ϕ(A(λ)P (λ) − 1)ψ ∈L−∞V (Rn; H; E, E) and ϕ(P (λ)A(λ) − 1)ψ ∈ L−∞

V (Rn; H;E,E).


2.5. Volterra Mellin calculus

Definition 2.5.1. Let E and E be Hilbert spaces. For µ ∈ R the spaces of (r, r′)-resp. r-dependent (classical) parameter-dependent Mellin symbols with respect tothe weight γ ∈ R and parameter-space Rn are defined as

MγSµ;ℓ(cl)(

(R+

)q × Rn × Γ 12−γ

;E, E) := C∞B (

(R+

)q, Sµ;ℓ

(cl)(Rn × Γ 1

2−γ;E, E))

for q = 1, 2.The spaces of (classical) Volterra Mellin symbols of order µ with respect to

the weight γ ∈ R are defined as

MγSµ;ℓV (cl)(

(R+

)q × Rn × H 12−γ

;E, E) := C∞B (

(R+

)q, Sµ;ℓ

V (cl)(Rn × H 1

2−γ;E, E))

for q = 1, 2.Analogously, we obtain the spaces of order −∞ with respect to the weight

γ ∈ R. All these spaces carry Frechet topologies in a canonical way.With the same conventions as in Definition 2.1.3 we also have the (Volterra)

Mellin symbol spaces when we deal with scales of Hilbert spaces instead of singleHilbert spaces only.

The operator of restriction of the half-plane H 12−γ

to the weight line Γ 12−γ

induces continuous embeddings

MγSµ;ℓV (cl)(

(R+

)q × Rn × H 12−γ

;E, E) → MγSµ;ℓ(cl)(

(R+

)q × Rn × Γ 12−γ

;E, E)

for q = 1, 2, see also Proposition 2.3.2.

Theorem 2.5.2. Let Ej and Ej be scales of Hilbert spaces, and E and E as inDefinition 2.1.6. Let (µk) ⊆ R such that µk −→


k∈N0

µk. Moreover,

let ak ∈ MγSµk;ℓ(R+ × Rn × Γ 1

2−γ; E , E). Then there exists a ∈ MγS

µ;ℓ(R+ ×

Rn × Γ 12−γ

; E , E) such that a ∼∞∑k=0

ak. The asymptotic expansion a is uniquely

determined modulo MγS−∞(R+ × Rn × Γ 1

2−γ; E , E).

If ak ∈ MγSµk;ℓV (R+ × Rn × H 1

2−γ; E , E) are given, then we find a ∈

MγSµ;ℓV (R+ × Rn × H 1

2−γ; E , E) such that a ∼

V

∞∑k=0

ak, and a is uniquely deter-

mined modulo MγS−∞V (R+ × Rn × H 1

2−γ; E , E).

If the sequence (µk)k∈N0 is given as µk = µ − k and the ak are classical(Volterra) Mellin symbols then also a is a classical (Volterra) Mellin symbol oforder µ.

Proof. This follows in the non-Volterra case as in the proof of Theorem 2.1.8 fromLemma 2.1.7. In the Volterra case we obtain the desired result analogous to theproof of Theorem 2.3.9 from Proposition 2.3.8.

Definition 2.5.3. Let E and E be Hilbert spaces, and µ ∈ R. With a Mellindouble-symbol a ∈ MγS

µ;ℓ(R+ × R+ × Rn × Γ 12−γ

;E, E) we associate a family


of Mellin pseudodifferential operators opγM (a)(ξ) ∈ L(Tγ(R+, E), Tγ(R+, E)) forξ ∈ Rn by means of the following Mellin oscillatory integral:

(opγM (a)(ξ)u)(r) : =1

2πi

∫

Γ 12−γ

∫

R+

( rr′

)−z

a(r, r′, ξ, z)u(r′)dr′

r′dz

=

∫

R

∫

R+

r′12−γ+iτ

a(r, rr′, ξ,1

2− γ + iτ)u(rr′)

dr′

r′dτ.

Taking into account the operator Sγ and its inverse from (1.1.1) and (1.1.2) wesee that we may write

opγM (a)(ξ) = S−1γ opt(aγ)(ξ)Sγ (2.5.1)

as operators on Tγ(R+, E), where the (Fourier) double-symbol aγ ∈ C∞b (R ×

R, Sµ;ℓ(Rn × R;E, E)) is given as

aγ(t, t′, ξ, τ) = a

(e−t, e−t

′

, ξ,1

2− γ + iτ

).

From (2.5.1) we thus see that the theory of Mellin pseudodifferential operators canbe carried over to some extent from the (standard) setting of operators based onthe Fourier transform.

Theorem 2.5.4. Let a ∈MγSµ;ℓ(cl)(R+ ×R+ ×Rn×Γ 1

2−γ;E, E). Then there exist

unique Mellin left- and right-symbols aL(r, ξ, z), aR(r′, ξ, z) ∈MγSµ;ℓ(cl)(R+ ×Rn ×

Γ 12−γ

;E, E) such that opγM (a)(ξ) = opγM (aL)(ξ) = opγM (aR)(ξ) as operators on

Tγ(R+, E).These symbols are given by the following Mellin oscillatory integrals:

aL(r, ξ, z) =

∫

R

∫

R+

siηa(r, sr, ξ, z + iη)ds

sdη,

aR(r′, ξ, z) =

∫

R

∫

R+

siηa(sr′, r′, ξ, z − iη)ds

sdη.

The mappings a 7→ aL and a 7→ aR are continuous. Moreover, we have the asymp-totic expansions

aL(r, ξ, z) ∼∞∑

k=0

1

k!Dkτ (−r′∂r′)ka(r, r′, ξ, z)|r′=r,

aR(r′, ξ, z) ∼∞∑

k=0

1

k!(−1)kDk

τ (−r∂r)ka(r, r′, ξ, z)|r=r′ .

If a ∈MγSµ;ℓV (cl)(R+ ×R+ ×Rn ×H 1

2−γ;E, E) then also aL, aR ∈MγS

µ;ℓV (cl)(R+ ×

Rn × H 12−γ

;E, E), and the mappings a 7→ aL and a 7→ aR are continuous with


respect to the topology of the Volterra Mellin symbol spaces. In this case we havethe asymptotic expansions

aL(r, ξ, z) ∼V

∞∑

k=0

1

k!∂kz (−r′∂r′)ka(r, r′, ξ, z)|r′=r,

aR(r′, ξ, z) ∼V

∞∑

k=0

1

k!(−1)k∂kz (−r∂r)ka(r, r′, ξ, z)|r=r′

in the Volterra sense.

Definition 2.5.5. For γ ∈ R define

MγLµ;ℓ(cl)(R+; Rn;E, E) := opγM (a)(ξ); a ∈MγS

µ;ℓ(cl)(R+ × Rn × Γ 1

2−γ;E, E),

MγLµ;ℓV (cl)(R+; Rn;E, E) := opγM (a)(ξ); a ∈MγS

µ;ℓV (cl)(R+ × Rn × H 1

2−γ;E, E).

In view of Theorem 2.5.4 we conclude that opγM (·)(ξ) provides an isomorphismbetween these spaces and the corresponding (left-) symbol spaces. Via that iso-morphism we carry over the topologies which turns the operator spaces into Frechetspaces.

Theorem 2.5.6. Let E, E and E be Hilbert spaces. Let a ∈MγSµ;ℓ(cl)(R+ × Rn ×

Γ 12−γ

; E, E) and b ∈ MγSµ′;ℓ(cl) (R+ × Rn × Γ 1

2−γ;E, E). Then the composition as

operators on Tγ(R+, E) may be written as

opγM (a)(ξ) opγM (b)(ξ) = opγM (a#b)(ξ)

with the Leibniz-product a#b ∈MγSµ+µ′;ℓ(cl) (R+×Rn×Γ 1

2−γ;E, E). More precisely,

the Leibniz-product is given by the Mellin oscillatory integral formula

a#b(r, ξ, z) =

∫

R

∫

R+

siηa(r, ξ, z + iη)b(rs, ξ, z)ds

sdη, (2.5.2)

and the following asymptotic expansion holds for a#b:

a#b ∼∞∑

k=0

1

k!(Dk

τa)((−r∂r)kb). (2.5.3)

The mapping (a, b) 7→ a#b is bilinear and continuous.

If a ∈ MγSµ;ℓV (cl)(R+ × Rn × H 1

2−γ; E, E) and b ∈ MγS

µ′;ℓV (cl)(R+ × Rn ×

H 12−γ

;E, E), then also a#b ∈MγSµ+µ′;ℓV (cl) (R+ ×Rn×H 1

2−γ;E, E), and the oscilla-

tory integral formula (2.5.2) is valid for z ∈ H 12−γ

, and the asymptotic expansion

(2.5.3) holds in the Volterra sense, i. e.,

a#b ∼V

∞∑

k=0

1

k!(∂kz a)((−r∂r)kb). (2.5.4)

In this case the mapping (a, b) 7→ a#b is bilinear and continuous within theVolterra Mellin symbol spaces.


Proof. The assertion follows from Theorem 2.5.4. Note that a(r, ξ, z)bR(r′, ξ, z) isa double-symbol for the composition, and the Leibniz-product is the associatedleft-symbol. This also implies the continuity of (a, b) 7→ a#b.

The oscillatory integral formula (2.5.2) necessarily holds in the preceding sit-uation, for it holds in the non-Volterra case without parameters, and by uniquenessof analytic continuation the formula is valid within the half-plane H 1

2−γ.

The asymptotic expansions (2.5.3) and (2.5.4) follow from (2.5.2) via Taylorexpansion.

Remark 2.5.7. The following pseudolocality property holds for Mellin operators:Let a(r, r′, ξ, z) ∈MγS

µ;ℓ(R+ × R+ × Rn × Γ 12−γ

;E, E) be a double-symbol

with a(r, r′, ξ, λ) ≡ 0 for∣∣ rr′ − 1

∣∣ < ε for a sufficiently small ε > 0. Then

opγM (a)(ξ) = opγM (c)(ξ) with a symbol c ∈ MγS−∞(R+ × Rn × Γ 1

2−γ;E, E). If

even a ∈MγSµ;ℓV (R+ ×R+ ×Rn×H 1

2−γ;E, E), then also c ∈MγS

−∞V (R+ ×Rn×

H 12−γ

;E, E).

In particular, if a(r, ξ, z) ∈ MγSµ;ℓ(R+ × Rn × Γ 1

2−γ;E, E) and ϕ, ψ ∈

C∞B (R+) such that dist(suppϕ, suppψ) > 0, then ϕ opγM (a)(ξ)ψ = opγM (aϕ,ψ)(ξ)

with a symbol aϕ,ψ ∈MγS−∞(R+×Rn×Γ 1

2−γ;E, E), and the mapping a 7−→ aϕ,ψ

is continuous. If even a(r, ξ, z) ∈ MγSµ;ℓV (R+ × Rn × H 1

2−γ;E, E), then also

aϕ,ψ ∈ MγS−∞V (R+ × Rn × H 1

2−γ;E, E). In this case the mapping a 7−→ aϕ,ψ

is continuous with respect to the Volterra Mellin symbol spaces.

Theorem 2.5.8. Let a ∈ MγSµ;ℓV (R+ × Rn × H 1

2−γ;E, E). Then opγM (a)(ξ) re-

stricts for every r0 ∈ R+ to a family of continuous operators

opγM (a)(ξ) : Tγ,0((0, r0), E) −→ Tγ,0((0, r0), E).

Proof. Without loss of generality assume n = 0. We may write(opγM (a)u

)(r) =

(M−1

γ,z→r′a(r, z)Mγ,r′→zu)|r′=r (1)

for u ∈ Tγ(R+, E). Now let u ∈ Tγ,0((0, r0), E) be given and r ∈ R+

fixed. In view of the Paley–Wiener characterizations (see Section 1.1) we haveMγu ∈ A(H 1

2−γ, E; r0). For a is a Volterra symbol by assumption we see that

a(r, z)(Mγu

)(z) may be regarded as an element of A(H 1

2−γ, E; r0), i. e., a acts as a

“multiplier” in the spaces A(H 12−γ

, ·; r0). Employing again the Paley–Wiener char-

acterizations we now conclude that M−1γ,z→r′a(r, z)Mγ,r′→zu ∈ Tγ,0((0, r0)r′ , E),

where the subscript r′ indicates that we consider the latter function space in thevariable r′. In particular, evaluation at r′ = r yields that (1) necessarily vanishesfor r > r0 which finishes the proof of the theorem.

Remark 2.5.9. Theorem 2.5.8 provides the motivation for the name “Volterra”symbols respectively operators:


If we regard the Mellin pseudodifferential operators as

opγM (a)(ξ) : C∞0 (R+) −→ C∞(R+,L(E, E)),

then we obtain for every r0 ∈ R+ as in Theorem 2.5.8 that(opγM (a)(ξ)u

)(r) ≡ 0

for r > r0, if u ∈ C∞0 (R+) such that u ≡ 0 for r > r0. In other words, the

operator-valued Schwartz kernel KopγM

(a)(ξ) ∈ D′(R+ × R+,L(E, E)) satisfies

suppKopγM (a)(ξ) ⊆ (r, r′) ∈ R+ × R+; r ≤ r′for all ξ ∈ Rn.

This gives the link to (classical) Volterra integral equations where the kernelis supported on one side of the diagonal only.

Continuity in Mellin Sobolev spaces.

Definition 2.5.10. Let E be a Hilbert space. For s, γ ∈ R define the MellinSobolev space Hs,γ(R+, E) to consist of all u ∈ T ′

γ (R+, E) such that Mγu is aregular distribution in S′(Γ 1

2−γ, E), and

‖u‖Hs,γ(R+,E) :=( 1

2πi

∫

Γ 12−γ

〈Im(z)〉2s‖Mγu(z)‖2E dz

) 12

<∞.

In case of E = C the space is suppressed from the notation.The operator Sγ from (1.1.1) provides an isomorphism Sγ : Hs,γ(R+, E) −→

Hs(R, E).For r0 ∈ R+ we define the space Hs,γ

0 ((0, r0], E) to consist of all u ∈Hs,γ(R+, E) such that supp(u) ⊆ (0, r0]. This is a closed subspace of Hs,γ(R+, E)and equals the closure of Tγ,0((0, r0), E) in Hs,γ(R+, E).

Theorem 2.5.11. Let E and E be Hilbert spaces. Let a ∈ MγSµ;ℓ(R+ × Rn ×

Γ 12−γ

;E, E) and s, ν ∈ R where ν ≥ µℓ . Then opγM (a)(ξ) extends for ξ ∈ Rn

by continuity to an operator opγM (a)(ξ) ∈ L(Hs,γ(R+, E),Hs−ν,γ(R+, E)), whichinduces a continuous embedding

MγLµ;ℓ(R+; Rn;E, E) →

Sµ(Rn;Hs,γ(R+, E),Hs−ν,γ(R+, E)) ν ≥ 0

Sµ−ν(Rn;Hs,γ(R+, E),Hs−ν,γ(R+, E)) ν ≤ 0

(2.5.5)into the space of operator-valued symbols in the Sobolev spaces.

Moreover, restriction of Volterra pseudodifferential operators to the Hs,γ0 -

spaces provides continuous mappings

MγLµ;ℓV (R+; Rn;E, E) −→

Sµ(Rn;Hs,γ

0 ((0, r0], E),Hs−ν,γ0 ((0, r0], E)) ν ≥ 0

Sµ−ν(Rn;Hs,γ0 ((0, r0], E),Hs−ν,γ

0 ((0, r0], E)) ν ≤ 0

for each r0 ∈ R+.


Remark 2.5.12. Employing relation (2.5.1) there is analogously a parameter-dependent pseudodifferential calculus with parameter-space Rn for operators basedon the Fourier transform, where the action in the covariable is carried out in the“Volterra”-covariable, i. e., the covariable which extends holomorphically into anupper or lower half-plane in C.

The analogue of Theorem 2.5.8 is valid within this calculus, which followsin the same way from Paley–Wiener characterizations as in the proof of Theorem2.5.8, but now with the Fourier transform involved. We will not state the detailsfor they are straightforward in view of the properties of the operator Sγ as well as(2.5.1) (see also Section 2.7).

2.6. Analytic Volterra Mellin calculus

Definition 2.6.1. Let E and E be Hilbert spaces. Moreover, let z = β + iτ ∈ C

be the splitting of z ∈ C in real and imaginary part. For µ ∈ R define the Frechetspaces

Sµ;ℓO(cl)(R

n × C;E, E) := A(C, Sµ(Rn;E, E)) ∩C∞(Rβ , Sµ;ℓ(cl)(R

n × Γβ;E, E)),

Sµ;ℓV,O(cl)(R

n×C;E, E) := Sµ;ℓO(cl)(R

n × C;E, E) ∩ Sµ;ℓV (cl)(R

n × H0;E, E)

with the induced topologies. Analogously, we define the corresponding symbolspaces when we deal with scales of Hilbert spaces.

Notation 2.6.2. For an interval ∅ 6= I ⊆ R we shall use the notation

ΓI := z ∈ C; Re(z) ∈ Ifor the strip in the complex plane over I.

Proposition 2.6.3. Let ∅ 6= I ⊆ R be an open interval and µ ∈ R. Let

ℓ∞locAµ;ℓ(cl)(E, E) := a ∈ A(ΓI , S

µ(Rn;E, E)); a|Γβ ∈ Sµ;ℓ(cl)(R

n × Γβ ;E, E)

locally uniformly for β ∈ I,C∞Aµ;ℓ

(cl)(E, E) := a ∈ A(ΓI , Sµ(Rn;E, E)); a ∈ C∞(Iβ , S

µ;ℓ(cl)(R

n × Γβ;E, E))endowed with their natural Frechet topologies. Observe that for I = R we recover

Sµ;ℓO(cl)(R

n×C;E, E) = C∞Aµ;ℓ(cl)(E, E).

a) The embedding ι : C∞Aµ;ℓ(cl)(E, E) → ℓ∞locAµ;ℓ

(cl)(E, E) is onto and provides an

isomorphism between these spaces.b) The complex derivative is a linear and continuous operator in the spaces ∂z :

ℓ∞locAµ;ℓ(cl)(E, E) → ℓ∞locAµ−ℓ;ℓ

(cl) (E, E).

c) Given a ∈ ℓ∞locAµ;ℓ(cl)(E, E), we have the following asymptotic expansion for a|Γβ0

in terms of a|Γβ for every β0, β ∈ I which depends smoothly on (β0, β) ∈ I×I:

a|Γβ0 ∼∞∑

k=0

(β0 − β)k

k!

(∂kz a

)|Γβ .


d) For arbitrary β ∈ R we have Sµ;ℓV,O(cl)(R

n×C;E, E) → Sµ;ℓV (cl)(R

n×Hβ;E, E).

If a ∈ Sµ;ℓV,O(cl)(R

n×C;E, E), then we have a|Hβ ∈ Sµ;ℓV (cl)(R

n×Hβ;E, E) as a

smooth function of β ∈ R, and the asymptotic expansion

a|Hβ0 ∼V

∞∑

k=0

(β0 − β)k

k!

(∂kz a

)|Hβ

is valid, which depends smoothly on (β0, β) ∈ R×R.e) For β ∈ I and µ ≥ µ′ the identity

ℓ∞locAµ;ℓ(E, E)∩Sµ′;ℓ

(cl) (Rn × Γβ ;E, E) = ℓ∞locAµ′;ℓ(cl) (E, E)

holds algebraically and topologically.f) For β ∈ R and µ ≥ µ′ the identity

Sµ;ℓV,O(Rn×C;E, E)∩Sµ

′;ℓV (cl)(R

n × Hβ;E, E) = Sµ′;ℓ

V,O(cl)(Rn×C;E, E)

holds algebraically and topologically.

From the expansions in c) and d) we see, that in the classical cases the homogeneousprincipal symbols of the restrictions do not depend on the particular weight line orhalf-plane.

Proposition 2.6.4. a) Let ∅ 6= I ⊆ R be an open interval. Moreover, let a ∈C∞B (R+, ℓ

∞locAµ;ℓ(E, E)) (cf. Proposition 2.6.3). Then for γ, γ′ ∈ R such that

12 − γ, 1

2 − γ′ ∈ I we have opγM (a)(ξ) = opγ′

M (a)(ξ) as operators on C∞0 (R+, E).

b) Let γ, δ ∈ R and a ∈ MγSµ;ℓ(R+ × Rn × Γ 1

2−γ;E, E). Then we have

opγM (a)(ξ)rδ = rδopγ−δM (T−δa)(ξ) as operators acting in Tγ−δ(R+, E) −→Tγ(R+, E), where T−δa ∈ Mγ−δS

µ;ℓ(R+ × Rn × Γ 12−γ+δ;E, E) is defined as(

T−δa)(r, ξ, 1

2 − γ + δ + iτ) := a(r, ξ, 12 − γ + iτ).

Proof. For the proof of a) note that we may write for u ∈ C∞0 (R+, E)

(opγM (a)(ξ)u

)(r) =

1

2πi

∫

Γ 12−γ

r−ζa(r, ξ, ζ)(Mu

)(ζ) dζ.

By Cauchy’s theorem we may change the line of integration from Γ 12−γ

to Γ 12−γ

′

which shows a).


We have to prove the asserted identity in b) only as operators on C∞0 (R+, E)

in view of the density. We may write for u ∈ C∞0 (R+, E)

(opγM (a)(ξ)

(rδu

))(r) =

1

2πi

∫

Γ 12−γ

r−ζa(r, ξ, ζ)(Mu

)(ζ + δ) dζ

=1

2πi

∫

Γ 12−γ+δ

r−(ζ−δ)(T−δa

)(r, ξ, ζ)

(Mu

)(ζ) dζ

= rδ(opγ−δM (T−δa)(ξ)u

)(r).

Definition 2.6.5. Let E and E be Hilbert spaces. For µ ∈ R the spaces of (r, r′)-resp. r-dependent (classical) parameter-dependent holomorphic Mellin symbolswith parameter-space Rn are defined as

MSµ;ℓO(cl)(

(R+

)q × Rn × C;E, E) := C∞B (

(R+

)q, Sµ;ℓ

O(cl)(Rn × C;E, E))

for q = 1, 2.The spaces of (classical) holomorphic Volterra Mellin symbols of order µ with

parameter-space Rn are defined as

MSµ;ℓV,O(cl)(

(R+

)q × Rn × C;E, E) := C∞B (

(R+

)q, Sµ;ℓ

V,O(cl)(Rn × C;E, E))

for q = 1, 2.Analogously, we obtain the spaces of order −∞. All these spaces carry Frechet

topologies in a canonical way.With the same conventions as in Definition 2.1.3 we also have the (Volterra)

Mellin symbol spaces when we deal with scales of Hilbert spaces instead of singleHilbert spaces only.

For every γ ∈ R the embeddings

MSµ;ℓO(cl)((R+)q × Rn × C;E, E) → MγS

µ;ℓ(cl)((R+)q × Rn × Γ 1

2−γ;E, E),

MSµ;ℓV,O(cl)((R+)q × Rn × C;E, E) → MγS

µ;ℓV (cl)((R+)q × Rn × H 1

2−γ;E, E)

are well-defined and continuous for q = 1, 2.

Definition 2.6.6. Let (µk) ⊆ R be a sequence of reals such that µk −→k→∞

−∞,

and µ := maxk∈N

µk. Moreover, let ak ∈ MSµk;ℓ(V,)O(R+ × Rn × C;E, E). A symbol

a ∈ MSµ;ℓ(V,)O(R+ × Rn × C;E, E) is called the asymptotic expansion of the ak, if

for every R ∈ R there is a k0 ∈ N such that for k > k0

a−k∑

j=1

aj ∈MSR;ℓ(V,)O(R+ × Rn × C;E, E).

The symbol a is uniquely determined modulo MS−∞(V,)O(R+ × Rn × C;E, E).


We shall again employ the notation a ∼(V )

∞∑j=1

aj.


Theorem 2.6.7. Let γ ∈ R and a ∈MSµ;ℓ(V,)O(cl)(R+ ×R+ ×Rn×C;E, E). Then

the Mellin left- and right-symbols aL(r, ξ, z), aR(r′, ξ, z) associated to the operator

opγM (a)(ξ) from Theorem 2.5.4 belong to MSµ;ℓ(V,)O(cl)(R+ × Rn × C;E, E) and do

not depend on the particular weight γ ∈ R. The oscillatory integral formulas foraL and aR in terms of a from Theorem 2.5.4 hold for z ∈ C, and the mappingsa 7→ aL and a 7→ aR are continuous.

Moreover, we have the asymptotic expansions in the sense of Definition 2.6.6:

aL(r, ξ, z) ∼(V )

∞∑

k=0

1

k!∂kz (−r′∂r′)ka(r, r′, ξ, z)|r′=r,

aR(r′, ξ, z) ∼(V )

∞∑

k=0

1

k!(−1)k∂kz (−r∂r)ka(r, r′, ξ, z)|r=r′.

Definition 2.6.8. Define

MOLµ;ℓ(cl)(R+; Rn;E, E) := opγM (a)(ξ); a ∈MSµ;ℓ

O(cl)(R+ × Rn × C;E, E),MV,OL

µ;ℓ(cl)(R+; Rn;E, E) := opγM (a)(ξ); a ∈MSµ;ℓ

V,O(cl)(R+ × Rn × C;E, E).In view of Theorem 2.6.7 opγM (·)(ξ) provides an isomorphism between the operatorspaces and the corresponding (left-) symbol spaces. Via that isomorphism we carryover the topologies which turns the operator spaces into Frechet spaces.

We do not refer to the particular weight γ ∈ R which is on the one handjustified by Theorem 2.6.7, and on the other hand by Proposition 2.6.4.

Theorem 2.6.9. Let E, E and E be Hilbert spaces, and let a ∈MSµ;ℓ(V,)O(cl)(R+×

Rn × C; E, E), as well as b ∈ MSµ′;ℓ

(V,)O(cl)(R+ × Rn × C;E, E). Then the Leibniz-

product a#b from Theorem 2.5.6 belongs to MSµ+µ′;ℓ(V,)O(cl)(R+ × Rn × C;E, E) and

is independent of the particular weight γ ∈ R. The oscillatory integral formula(2.5.2) for a#b in terms of a and b holds for z ∈ C, and the mapping (a, b) 7→ a#bis bilinear and continuous.

The following asymptotic expansion holds for a#b in the sense of Definition2.6.6:

a#b ∼(V )

∞∑

k=0

1

k!(∂kz a)((−r∂r)kb). (2.6.1)

Moreover, we have the following formula for the derivatives of the Leibniz-product:

∂kr(a#b

)=

k∑

j=0

(k

j

)(T−(k−j)∂

jra

)#

(∂k−jr b

), (2.6.2)

where T denotes the translation operator for functions in the complex plane.


Definition 2.6.10. Let opγM (a)(ξ) ∈MOLµ;ℓ(R+; Rn; E, E). For k ∈ N0 we define

the conormal symbol of order -k via

σ−kM

(opγM (a)(ξ)

)(ξ, z) :=

1

k!

(∂kr a

)(0, ξ, z). (2.6.3)

The conormal symbol of order 0 is also called conormal symbol simply.Let opγM (b)(ξ) ∈ MOL

µ′;ℓ(R+; Rn;E, E). Then we obtain from (2.6.2) thefollowing formula for the conormal symbols of the composition

σ−kM

(opγM (a#b)(ξ)

)=

∑

p+q=k

T−qσ−pM

(opγM (a)(ξ)

)σ−qM

(opγM (b)(ξ)

), (2.6.4)

where T denotes the translation operator for functions in the complex plane.

Remark 2.6.11. By Theorem 2.6.7 the following pseudolocality property of thecalculus is valid:

Let a(r, r′, ξ, z) ∈MSµ;ℓ(V,)O(R+×R+×Rn×C;E, E), such that a(r, r′, ξ, λ) ≡ 0

for∣∣ rr′ − 1

∣∣ < ε for a sufficiently small ε > 0. Then opγM (a)(ξ) = opγM (c)(ξ) with

a symbol c ∈MS−∞(V,)O(R+ × Rn × C;E, E).

In particular, if a(r, ξ, z) ∈MSµ;ℓ(V,)O(R+×Rn×C;E, E), and ϕ, ψ ∈ C∞

B (R+)

such that dist(suppϕ, suppψ) > 0, then ϕ opγM (a)(ξ)ψ = opγM (aϕ,ψ)(ξ) with a

symbol aϕ,ψ ∈MS−∞(V,)O(R+×Rn×C;E, E). The mapping a 7−→ aϕ,ψ is continuous.

The Mellin kernel cut-off operator and asymptotic expansion.

Definition 2.6.12. Let E and E be Hilbert spaces. Define the Mellin kernel cut-off operator with respect to the weight γ ∈ R by means of the Mellin oscillatoryintegral

(Hγ(ϕ)a

)(ξ, z) :=

∫

R

∞∫

0

riτϕ(r)a(ξ, z − iτ)dr

rdτ (2.6.5)

for (ξ, z) ∈ Rn × Γ 12−γ

and ϕ ∈ C∞B (R+), a ∈ Sµ;ℓ(Rn × Γ 1

2−γ;E, E).

Note that we may rewrite (2.6.5) as

(Hγ(ϕ)a

)(ξ,

1

2− γ + iτ) =

(H

(S 1

2ϕ)aγ

)(ξ, τ)

with the Fourier kernel cut-off operator H as introduced in Definition 2.3.5, thetransformation S 1

2: C∞

B (R+) −→ C∞b (R) from (1.1.1), and aγ(ξ, τ) := a(ξ, 1

2 −γ + iτ).

Analogous notions apply in case of scales of Hilbert spaces involved.

Theorem 2.6.13. Let Ej and Ej be scales of Hilbert spaces as in Definition2.1.3. We again use the abbreviations

E := ind-limj∈N


Ej .


The Mellin kernel cut-off operator with respect to the weight γ ∈ R acts as acontinuous bilinear mapping in the spaces

Hγ : C∞B (R+)×Sµ;ℓ

(cl)(Rn × Γ 1

2−γ; E , E) −→ Sµ;ℓ

(cl)(Rn × Γ 1

2−γ; E , E).

It restricts to continuous bilinear mappings in the spaces

Hγ :

C∞B (R+)×Sµ;ℓ

V (cl)(Rn × H 1

2−γ; E , E) −→ Sµ;ℓ

V (cl)(Rn × H 1

2−γ; E , E)

C∞0 (R+)×Sµ;ℓ

(cl)(Rn × Γ 1

2−γ; E , E) −→ Sµ;ℓ

O(cl)(Rn × C; E , E)

C∞0 (R+)×Sµ;ℓ

V (cl)(Rn × H 1

2−γ; E , E) −→ Sµ;ℓ

V,O(cl)(Rn × C; E , E).

The following asymptotic expansion holds for Hγ(ϕ)a ∈ Sµ;ℓ(Rn × Γ 12−γ

; E , E) in

terms of ϕ ∈ C∞B (R+) and a ∈ Sµ;ℓ(Rn × Γ 1

2−γ; E , E):

Hγ(ϕ)a ∼∞∑

k=0

1

k!(r∂r)

kϕ(r)|r=1 ·Dkτa.

In case of Volterra symbols we obtain

Hγ(ϕ)a ∼V

∞∑

k=0

1

k!(r∂r)

kϕ(r)|r=1 · ∂kz a.

If ϕ ∈ C∞0 (R+) and a ∈ Sµ;ℓ(Rn × Γ 1

2−γ; E , E) we have for every δ ∈ R the

following asymptotic expansion of Hγ(ϕ)a|Γ 12−γ−δ

∈ Sµ;ℓ(Rn × Γ 12−γ−δ

; E , E) in

terms of ϕ and a:

Hγ(ϕ)a|Γ 12−γ−δ

∼∞∑

k=0

1

k!(r∂r)

kr−δϕ(r)|r=1 ·Dkτa.

If ψ ∈ C∞0 (R+) such that ψ ≡ 1 near r = 1, then the operator I − Hγ(ψ) is

continuous in the spaces

I −Hγ(ψ) :

Sµ;ℓ(Rn × Γ 1

2−γ; E , E) −→ S−∞(Rn × Γ 1

2−γ; E , E)

Sµ;ℓV (Rn × H 1

2−γ; E , E) −→ S−∞

V (Rn × H 12−γ

; E , E).

Theorem 2.6.14. Let (µk) ⊆ R such that µk −→k→∞

−∞ and µ := maxk∈N0

µk. More-

over, let ak ∈ MSµk;ℓ(V,)O(R+ × Rn × C; E , E). Then there exists a ∈MSµ;ℓ(V,)O(R+ ×

Rn×C; E , E) such that a ∼(V )

∞∑k=0

ak in the sense of Definition 2.6.6. The asymptotic

expansion a is uniquely determined modulo MS−∞(V,)O(R+ × Rn × C; E , E).

If the sequence (µk)k∈N0 is given as µk = µ − k and the ak are classical(Volterra) Mellin symbols then also a is a classical (Volterra) Mellin symbol oforder µ.


Proof. According to Lemma 2.1.7 and Theorem 2.1.8, or Proposition 2.3.8 andTheorem 2.3.9, respectively, we first obtain a symbol

a ∈C∞B (R+, S

µ;ℓ(Rn × Γ0; E , E))

C∞B (R+, S

µ;ℓV (Rn × H0; E , E))

such that a ∼∞∑k=0

ak|Γ0 , respectively a ∼V

∞∑k=0

ak|H0 . Now define a := H 12(ψ)a with

the Mellin kernel cut-off operator H 12, and a function ψ ∈ C∞

0 (R+) such that

ψ ≡ 1 near r = 1. From Theorem 2.6.13 we now obtain that a ∈ MSµ;ℓ(V,)O(R+ ×

Rn × C; E , E), and moreover a|Γ0 ∼ a resp. a|H0 ∼Va. From Proposition 2.6.3 we

now conclude that indeed a ∼(V )

∞∑k=0

ak in the sense of Definition 2.6.6 as asserted.

In the classical case we have a and consequently also a as classical symbols.

Degenerate symbols and Mellin quantization.

Definition 2.6.15. Let E and E be Hilbert spaces. For ϕ ∈ C∞0 (R+) and a ∈

Sµ;ℓ(Rn × R;E, E) define

Q(ϕ, a)(ξ, z) :=

∫

R

∫

R

e−isηeiηs−zϕ(s)a(ξ, η) ds dη (2.6.6)

for (ξ, z) ∈ Rn × C.Moreover, for every γ ∈ R we define for ψ ∈ C∞

0 (R+) and a ∈ Sµ;ℓ(Rn ×Γ 1

2−γ;E, E)

Qγ(ψ, a)(ξ, z) :=1

2πi

∫

Γ 12−γ

∞∫

0

sζei(s−1)zψ(s)a(ξ, ζ)ds

sdζ (2.6.7)

for (ξ, z) ∈ Rn × C.If ϕ ≡ 1 near r = 1 respectively ψ ≡ 1 near r = 1 we simply write

Q(ϕ, a) = Q(a) and Qγ(ψ, a) = Qγ(a), respectively. The mapping Q is called

Mellin quantization, Qγ is called inverse Mellin quantization with respect to theweight γ ∈ R.

Theorem 2.6.16. Let Ej and Ej be scales of Hilbert spaces, and E and E asbefore.

a) The operator Q from (2.6.6) provides continuous bilinear mappings

Q :

C∞

0 (R+)×Sµ;ℓ(cl)(R

n × R; E , E) −→ Sµ;ℓO(cl)(R

n×C; E , E)

C∞0 (R+)×Sµ;ℓ

V (cl)(Rn × H; E , E) −→ Sµ;ℓ

V,O(cl)(Rn×C; E , E).

Moreover, there are universal coefficients (ck,j(ϕ, γ)) depending neither on anor on the Hilbert spaces, but only on

(∂νrϕ

)(1); ν ∈ N0 and γ ∈ R, such


that the following asymptotic expansion holds for Q(ϕ, a)|Γ 12−γ

, respectively

Q(ϕ, a)|H 12−γ

, in terms of a:

Q(ϕ, a)(ξ,1

2− γ + iτ) ∼

(V )ϕ(1)a(ξ,−τ) +

∞∑

k=1

k∑

j=0

ck,j(ϕ, γ)(−τ)j(∂k+jτ a

)(ξ,−τ)

(2.6.8)for τ ∈ R, respectively τ ∈ H−.

b) The operator Qγ from (2.6.7) provides continuous bilinear mappings

Qγ :

C∞

0 (R+)×Sµ;ℓ(cl)(R

n × Γ 12−γ

; E , E) −→ Sµ;ℓiO(cl)(R

n×C; E , E)

C∞0 (R+)×Sµ;ℓ

V (cl)(Rn × H 1

2−γ; E , E) −→ Sµ;ℓ

V,iO(cl)(Rn×C; E , E).

The spaces in the image are given by means of the isomorphism

Sµ;ℓ(V,)iO(cl)(R

n×C; E , E) ∋ a(ξ, z) 7−→ a(ξ,−iz) ∈ Sµ;ℓ(V,)O(cl)(R

n×C; E , E).

Moreover, there are universal coefficients (dk,j(ψ, γ)) depending neither on anor on the Hilbert spaces, but only on

(∂νrψ

)(1); ν ∈ N0 and γ ∈ R,

such that the following asymptotic expansion holds for Qγ(ψ, a)|R, respectively

Qγ(ψ, a)|H, in terms of a:

Qγ(ψ, a)(ξ, τ) ∼(V )

ψ(1)a(ξ,1

2− γ − iτ)

+∞∑

k=1

k∑

j=0

dk,j(ψ, γ)(−iτ)j(∂k+jτ a

)(ξ,

1

2− γ − iτ)

(2.6.9)

for τ ∈ R, respectively τ ∈ H.c) For ϕ, ψ ∈ C∞

0 (R+) such that ϕ ≡ 1 and ψ ≡ 1 near r = 1 we have

Q(Qγ(a)) − a ∈S−∞(Rn×Γ 1

2−γ; E , E)

S−∞V (Rn×H 1

2−γ; E , E),

Qγ(Q(a)) − a ∈S−∞(Rn×R; E , E)

S−∞V (Rn×H; E , E).

Remark 2.6.17. Let ϕ, ψ ∈ C∞0 (R+) such that ϕ ≡ 1 and ψ ≡ 1 near r = 1.

By Theorem 2.6.16 the mappings Q : a 7→ Q(a) and Qγ : a 7→ Qγ(a) provide


isomorphisms

Q :

Sµ;ℓ(cl)(R

n × R;E , E)/S−∞(Rn × R; E , E)

−→ Sµ;ℓO(cl)(R

n×C; E , E)/S−∞O (Rn×C; E , E)

Sµ;ℓV (cl)(R

n × H;E , E)/S−∞V (Rn × H; E , E)

−→ Sµ;ℓV,O(cl)(R

n×C; E , E)/S−∞V,O (Rn×C; E , E),

Qγ :

Sµ;ℓ(cl)(R

n × Γ 12−γ

;E , E)/S−∞(Rn × Γ 12−γ

; E , E)

−→ Sµ;ℓiO(cl)(R

n×C; E , E)/S−∞iO (Rn×C; E , E)

Sµ;ℓV (cl)(R

n × H 12−γ

;E , E)/S−∞V (Rn × H 1

2−γ; E , E)

−→ Sµ;ℓV,iO(cl)(R

n×C; E , E)/S−∞V,iO(Rn×C; E , E).

On the level of quotient spaces, Q and Qγ are independent of ϕ and ψ, respectively,

and we have Qγ = Q−1 (according to part c) of Theorem 2.6.16).

Theorem 2.6.18. Let E and E be Hilbert spaces, and let ϕ, ψ ∈ C∞0 (R+) be fixed

such that ϕ ≡ 1 and ψ ≡ 1 near r = 1.

a) Let a ∈ C∞(R+, Sµ;ℓ(Rn × R;E, E)), and define a(r, ξ, τ) := a(r, ξ, rτ). Then

we have for every γ ∈ R

opr(a)(ξ) − opγM (Q(a))(ξ) = opr((1 − ϕ)

(r′r

)a)(ξ)

as operators in C∞0 (R+, E) −→ C∞(R+, E).

b) Let a ∈ C∞(R+, Sµ;ℓ(Rn×Γ 1

2−γ;E, E)), and define a(r, ξ, τ) := Qγ(a)(r, ξ, rτ).

Then we have

opγM (a)(ξ) − opr(a)(ξ) = opγM ((1 − ψ)(r′r

)a)(ξ)

as operators in C∞0 (R+, E) −→ C∞(R+, E).

Remark 2.6.19. Theorem 2.6.18 gives the explanation for the name “Mellin quan-tization” for the operator Q, and “inverse Mellin quantization” for Q. Togetherwith Theorem 2.6.16 it follows, that modulo “smoothing” (Volterra) operators weobtain isomorphisms between Fourier pseudodifferential operators with degener-ate (Volterra) symbols on the half-axis and (Volterra) Mellin pseudodifferentialoperators.

Note in particular, that if in a) or b) the dependence of the symbol a on

r ∈ R+ is C∞B (R+, ·), then so is also the dependence of Q(a) or Q(a), respectively.

2.7. Volterra Fourier operators with global weight conditions

Remark 2.7.1. In this section we briefly recall the elements of a parameter-dependent pseudodifferential calculus based on the Fourier transform, where theaction is carried out with respect to the Volterra covariable, and the symbols glob-ally satisfy weighted estimates in the variable. In view of the considerations fromSection 2.5, note in particular relation (2.5.1), these are easily obtained together


with the general theory of such global operators, see, e. g., Cordes [10], Dorschfeldt,Grieme, and Schulze [11], Parenti [45], Schrohe [55], or Seiler [64].

Definition 2.7.2. Let E and E be Hilbert spaces, and let again H be the upperhalf-plane in C, and µ, 1, 2 ∈ R. The spaces of globally weighted (Volterra)double- resp. left-/right- symbols with parameter-space Rn are defined as

Sµ,1,2;ℓ(cl) (R × R × Rn × R;E, E) := S1,2(R × R, Sµ;ℓ

(cl)(Rn × R;E, E)),

Sµ,1;ℓ(cl) (R × Rn × R;E, E) := S1(R, Sµ;ℓ(cl)(R

n × R;E, E)),

Sµ,1,2;ℓV (cl) (R × R × Rn × H;E, E) := S1,2(R × R, Sµ;ℓV (cl)(R

n × H;E, E)),

Sµ,1;ℓV (cl) (R × Rn × H;E, E) := S1(R, Sµ;ℓ

V (cl)(Rn × H;E, E)).

With a symbol a ∈ Sµ,1,2;ℓ(R × R × Rn × R;E, E) we associate a family ofpseudodifferential operators acting as continuous operators

opr(a)(ξ) : S(R, E) −→ S(R, E)

for ξ ∈ Rn as in Section 2.2. The corresponding operator spaces are denoted asfollows:

Lµ,1;ℓ(cl) (R; Rn;E, E) := opr(a)(ξ); a ∈ Sµ,1;ℓ

(cl) (R × Rn × R;E, E),Lµ,1;ℓV (cl) (R; Rn;E, E) := opr(a)(ξ); a ∈ Sµ,1;ℓ

V (cl) (R × Rn × H;E, E).As before, Theorem 2.7.4 below guarantees that these spaces are well-defined inthe sense, that the action applied to left-symbols only already gives the full spaceof operators, and by means of the uniqueness of the left-symbol for the action ofthe operator we have the canonical isomorphism between symbols and operators,which induces a Frechet topology on the operator spaces.

Remark 2.7.3. In the sequel, the asymptotic expansions are to be understood inthe following sense:Let (µk), (k) ⊆ R be sequences such that µk, k −→

k→∞−∞, and µ := max

k∈N

µk as

well as := maxk∈N

k. Moreover, let

ak ∈Sµk,k;ℓ(R × Rn × R;E, E)

Sµk,k;ℓV (R × Rn × H;E, E),

a ∈Sµ,;ℓ(R × Rn × R;E, E)

Sµ,;ℓV (R × Rn × H;E, E).

We write a ∼(V )

∞∑j=1

aj if for every R ∈ R there is a k0 ∈ N such that for k > k0

a−k∑

j=1

aj ∈SR,R;ℓ(R × Rn × R;E, E)

SR,R;ℓV (R × Rn × H;E, E).


Theorem 2.7.4. Let a ∈ Sµ,1,2;ℓ(R × R × Rn × R;E, E). Then there exist

unique left- and right-symbols aL(r, ξ, τ), aR(r′, ξ, τ) ∈ Sµ,1+2;ℓ(R×Rn×R;E, E)such that opr(a)(ξ) = opr(aL)(ξ) = opr(aR)(ξ) as operators on S(R, E). Thesesymbols are given by means of oscillatory integral formulas analogous to that ofTheorem 2.2.2. The class of Volterra symbols remains preserved, i. e., if a ∈Sµ,1,2;ℓV (R×R×Rn×H;E, E), then also aL, aR ∈ Sµ,1+2;ℓ

V (R×Rn×H;E, E).More precisely, the mappings a 7−→ aL, aR are well-defined and continuous

in the spaces

Sµ,1,2;ℓ(cl) (R × R × Rn × R;E, E)

Sµ,1,2;ℓV (cl) (R × R × Rn × H;E, E)

−→

Sµ,1+2;ℓ

(cl) (R × Rn × R;E, E)

Sµ,1+2;ℓV (cl) (R × Rn × H;E, E).

Moreover, we have the asymptotic expansions

aL(r, ξ, τ) ∼(V )

∞∑

k=0

1

k!∂kτD

kr′a(r, r

′, ξ, τ)|r′=r,

aR(r′, ξ, τ) ∼(V )

∞∑

k=0

1

k!(−1)k∂kτD

kra(r, r

′, ξ, τ)|r=r′ .

Theorem 2.7.5. Let E, E and E be Hilbert spaces. Let

a ∈Sµ,;ℓ(cl) (R × Rn × R; E, E)

Sµ,;ℓV (cl)(R × Rn × H; E, E),

b ∈Sµ

′,′;ℓ(cl) (R × Rn × R;E, E)

Sµ′,′;ℓ

V (cl) (R × Rn × H;E, E),

and A(ξ) = opr(a)(ξ), B(ξ) = opr(b)(ξ). Then the composition as operators onS(R, E) is given as A(ξ)B(ξ) = opr(a#b)(ξ) with the Leibniz-product

a#b ∈Sµ+µ′,+′;ℓ

(cl) (R × Rn × R;E, E)

Sµ+µ′,+′;ℓV (cl) (R × Rn × H;E, E).

of the symbols a and b. The analogue of the oscillatory integral formula (2.2.1)for the Leibniz-product from Theorem 2.2.4 is valid, and we have the asymptoticexpansion

a#b ∼(V )

∞∑

k=0

1

k!(∂kτ a)(D

kr b).

Remark 2.7.6. From Theorem 2.7.4 we obtain the following pseudolocality prop-erty of this calculus:

Let

a(r, r′, ξ, τ) ∈Sµ,1,2;ℓ(R × R × Rn × R;E, E)

Sµ,1,2;ℓV (R × R × Rn × H;E, E)


such that a(r, r′, ξ, τ) ≡ 0 for |r − r′| < ε for a sufficiently small ε > 0. Then

opr(a)(ξ) ∈L−∞,−∞(R; Rn;E, E)

L−∞,−∞V (R; Rn;E, E).

Proposition 2.7.7. Let a ∈ Sµ,;ℓV (R × Rn × H;E, E). Then opr(a)(ξ) restrictsfor every r0 ∈ R to a family of continuous operators

opr(a)(ξ) : S0((−∞, r0), E) −→ S0((−∞, r0), E).

Proof. This follows analogously to Theorem 2.5.8.

Notation 2.7.8. Let [·] : R −→ R+ be a smoothed norm function, i. e., [·] ∈C∞(R,R+), and [r] ≡ |r| for |r| sufficiently large. Note that [·] ∈ S(R) for every ∈ R.

Theorem 2.7.9. Let E0, E,E1 and E0, E, E1 be Hilbert triples, and A(ξ) =

opr(a)(ξ) with a ∈ Sµ,;ℓ(cl) (R × Rn × R;E0, E0). Then the formal adjoint operators

A(ξ)(∗),δ with respect to the [·]−δL2-inner product belong to Lµ,;ℓ(cl) (R; Rn; E1, E1).

More precisely, for u ∈ S(R, E0) and v ∈ S(R, E1) we have∫

R

〈(A(ξ)u

)(r), v(r)〉E [r]2δ dr =

∫

R

〈u(r),(A(ξ)(∗),δv

)(r)〉E [r]2δ dr

with A(ξ)(∗),δ = opr(a(∗),δ)(ξ), where a(∗),δ ∈ Sµ,;ℓ(cl) (R × Rn × R; E1, E1) is given

asa(∗),δ =

([r]−2δ

(a(r′, ξ, τ)

)∗[r′]2δ

)L.

We have the asymptotic expansion

a(∗),δ(r, ξ, τ) ∼∞∑

k=0

∑

p+q=k

1

p!q!

([r]−2δDp

r [r]2δ

)(∂kτD

qr(a(r, ξ, τ))

∗).

Definition 2.7.10. Let E be a Hilbert space. For s, δ ∈ R define

Hs,δ(R, E) := 〈r〉−δHs(R, E)

with the Sobolev spaceHs(R, E), see Definition 2.2.12. This space is endowed withthe induced scalar product which turns it into a Hilbert space.

Moreover, for r0 ∈ R let Hs,δ0 ((−∞, r0], E) denote the closed subspace of

all u ∈ Hs,δ(R, E) such that supp(u) ⊆ (−∞, r0], which equals the closure ofS0((−∞, r0), E) in Hs,δ(R, E).

Theorem 2.7.11. Let E and E be Hilbert spaces. Moreover, let a ∈ Sµ,;ℓ(R ×Rn × R;E, E), and s, δ, ν, δ′ ∈ R where ν ≥ µ

ℓ and δ′ ≥ .Then opr(a)(ξ) extends for ξ ∈ Rn by continuity to an operator opr(a)(ξ) ∈

L(Hs,δ(R, E), Hs−ν,δ−δ′(R, E)), which induces a continuous embedding

Lµ,;ℓ(R; Rn;E, E) →Sµ(Rn;Hs,δ(R, E), Hs−ν,δ−δ′ (R, E)) ν ≥ 0

Sµ−ν(Rn;Hs,δ(R, E), Hs−ν,δ−δ′ (R, E)) ν ≤ 0



Moreover, restriction of Volterra pseudodifferential operators on the Hs,δ0 -

spaces provides continuous mappings

Lµ,;ℓV (R; Rn;E, E) −→

Sµ(Rn;Hs,δ0 ((−∞, r0], E), Hs−ν,δ−δ′

0 ((−∞, r0], E))

if ν ≥ 0

Sµ−ν(Rn;Hs,δ0 ((−∞, r0], E), Hs−ν,δ−δ′

0 ((−∞, r0], E))

if ν ≤ 0

for each r0 ∈ R.

Chapter 3. Parameter-dependent Volterra calculus on a closed

manifold

3.1. Anisotropic parameter-dependent operators

Notation 3.1.1. Let X be a closed manifold of dimension dimX = n, and E andF be complex vector bundles over X of dimensions N− and N+, respectively. Alocal chart will be denoted as a tuple (κ,Ω, U) (or simply κ), where Ω ⊆ X andU ⊆ Rn are open subsets and κ : Ω → U is a diffeomorphism. We will throughoutassume that the bundles are trivial over Ω. The transition matrices of the fibres(local trivializations of the bundles) are suppressed from the notation.

On X we fix the following data:

• A finite open covering Ωj ; j = 1, . . . , N of X , where Ωj ⊆ Ωj ⋐ Ωj withsuitable coordinate neighbourhoods (κj,Ωj , Uj); j = 1, . . . , N such thatE and F are trivial over Ωj .

• A subordinated C∞-partition of unity ϕjj=1,...,N , i. e., ϕj ∈ C∞0 (Ωj)

with 0 ≤ ϕj ≤ 1 andN∑j=1

ϕj ≡ 1.

• Suitable functions ψj ∈ C∞0 (Ωj) and θj ∈ C∞

0 (Ωj) with ψj ≡ 1 in a

neighbourhood of supp(ϕj) and θj ≡ 1 on Ωj for j = 1, . . . , N .• A Riemannian metric on X .• Hermitean inner products on E and F .

The Riemannian metric and the Hermitean inner products on the bundles are fixedin order to avoid inconveniences what the measure on X and the Hilbert spacestructure of L2(X,E) are concerned. Alternatively, we could also fix a positivesection in the density bundle over X replacing the Riemannian metric, or considersections in the 1

2 -density bundle instead of working with functions.Pull-backs and push-forwards of distributions and operators with respect to

a chart κ will be denoted by κ∗ and κ∗, respectively. Note once more that localtrivializations of the bundles are suppressed from the notation.


Definition 3.1.2. a) We define L−∞(X ; Rq;E,F ) := S(Rq, L−∞(X ;E,F )),where L−∞(X ;E,F ) denotes the space of smoothing pseudodifferential op-erators on X acting from C∞(X,E) to C∞(X,F ). This space carries a naturalFrechet topology, and it is characterized as

L−∞(X ; Rq;E,F ) = S(Rq,⋂

s,t∈R

L(Hs(X,E), Ht(X,F )))

=⋂

s,t∈R

S(Rq ,L(Hs(X,E), Ht(X,F )))

= S−∞(Rq; ind-lims∈R

Hs(X,E), proj-limt∈R

Ht(X,F )).

In other words, the elements A(λ) ∈ L−∞(X ; Rq;E,F ) are precisely thoseoperators having integral kernels k(x, y, λ) ∈ S(Rqλ, C

∞(Xx×Xy, F ⊠ E∗)).

b) The space Lµ;ℓ(cl)(X ; Rq;E,F ) consists of all families of operators A(λ) :

C∞(X,E) → C∞(X,F ) with the following properties:i) For all functions ϕ, ψ ∈ C∞(X) that are supported in the same coor-

dinate neighbourhood (κ,Ω, U) the push-forward κ∗(ϕA(λ)ψ) belongs to

Lµ;ℓcomp (cl)(U ; Rq; CN− ,CN+).

Note that ϕA(λ)ψ acts as an operator from C∞0 (Ω, E) to C∞

0 (Ω, F ), andconsequently κ∗(ϕA(λ)ψ) : C∞

0 (U,CN−) → C∞0 (U,CN+). The condition

is, that this operator should belong to Lµ;ℓcomp (cl)(U ; Rq; CN− ,CN+) →

L(C∞0 (U,CN−), C∞

0 (U,CN+)).ii) For all ϕ, ψ ∈ C∞(X) with disjoint support the operator ϕA(λ)ψ belongs

to L−∞(X ; Rq;E,F ).

We endow the space Lµ;ℓ(cl)(X ; Rq;E,F ) with the projective topology with respect

to the mappings A(λ) 7−→κ∗(ϕA(λ)ψ) ∈ Lµ;ℓ

comp(cl)(U ; Rq; CN− ,CN+) for ϕ, ψ supported in (κ,Ω, U)

ϕA(λ)ψ ∈ L−∞(X ; Rq;E,F ) for suppϕ ∩ suppψ = ∅.

Theorem 3.1.3. a) Let (κ,Ω, U) be a chart. Then the pull-back κ∗(A(λ)) of ev-

ery compactly supported operator family A(λ) ∈ Lµ;ℓcomp (cl)(U ; Rq; CN− ,CN+)

belongs to Lµ;ℓ(cl)(X ; Rq;E,F ). Moreover, this induces a continuous linear map-

ping

Lµ;ℓcomp (cl)(U ; Rq; CN− ,CN+) ∋ A(λ) 7−→ κ∗(A(λ)) ∈ Lµ;ℓ

(cl)(X ; Rq;E,F ).

Note: A(λ) acts as a continuous operator C∞0 (U,CN−) → C∞

0 (U,CN+), andthere exists a function ϕ ∈ C∞

0 (U) with ϕA(λ)ϕ = A(λ). Therefore, thepull-back κ∗(A(λ)) = κ∗(ϕ)κ∗(A(λ))κ∗(ϕ) is defined as a continuous opera-tor C∞

0 (Ω, E) → C∞0 (Ω, F ), which extends by means of the latter identity to an

operator C∞(X,E) → C∞(X,F ). The assertion now is, that κ∗(A(λ)) belongs

to Lµ;ℓ(cl)(X ; Rq;E,F ) in this sense.


b) Let (κ,Ω, U) be a chart, ϕ, ψ ∈ C∞0 (Ω), and A(λ) ∈ Lµ;ℓ

(cl)(Rn; Rq; CN− ,CN+).

Then the operator ϕκ∗(A(λ))ψ belongs to Lµ;ℓ(cl)(X ; Rq;E,F ), and the mapping

Lµ;ℓ(cl)(R

n; Rq; CN− ,CN+) ∋ A(λ) 7−→ ϕκ∗(A(λ))ψ ∈ Lµ;ℓ(cl)(X ; Rq;E,F )

is continuous. Note that for short we write κ∗(A(λ)) for κ∗(A(λ)|C∞

0 (U,CN−)).

c) Let A(λ) ∈ Lµ;ℓ(cl)(X ; Rq;E,F ). We may write

A(λ) =N∑

j=1

ϕjA(λ)ψj +( N∑

j=1

ϕjA(λ)(1 − ψj))

︸︷︷︸=:K(λ)∈L−∞(X;Rq ;E,F )

=N∑

j=1

ϕj(θjA(λ)θj)ψj +K(λ).

The mapping

Lµ;ℓ(cl)(X ; Rq;E,F ) ∋ A(λ) 7−→ (κ1,∗(θ1A(λ)θ1), . . . , κN,∗(θNA(λ)θN );K(λ))

∈N×j=1

Lµ;ℓcomp (cl)(Uj ; R

q; CN− ,CN+) × L−∞(X ; Rq;E,F )

(3.1.1)

is continuous. Conversely, the mapping

N×j=1

Lµ;ℓ(cl)(R

n; Rq;CN− ,CN+) × L−∞(X ; Rq;E,F ) ∋ (A1(λ), . . . , AN (λ);K(λ))

7−→N∑

j=1

ϕjκ∗j (Aj(λ))ψj +K(λ) ∈ Lµ;ℓ

(cl)(X ; Rq;E,F )

(3.1.2)

is continuous, and (3.1.1) is right-inverse to (3.1.2).

d) Lµ;ℓ(cl)(X ; Rq;E,F ) is a Frechet space, and the embedding Lµ;ℓ

(cl)(X ; Rq;E,F ) →C∞(Rq, Lµ(X ;E,F )) is continuous.

For β ∈ Nq0 the operator ∂βλ : Lµ;ℓ

(cl)(X ; Rq;E,F ) → Lµ−ℓ|β|;ℓ(cl) (X ; Rq;E,F ) is

continuous.

Proof. a) follows from Theorem 2.2.15 and Remark 2.2.5. Note that the transitionbetween different trivializations of the vector bundles results in “conjugation”with the transition matrices in the local representations for the operator. To proveb), choose a function θ ∈ C∞

0 (Ω) with θψ = ψ and θϕ = ϕ. Then we haveϕκ∗(A(λ))ψ = ϕκ∗(κ∗θA(λ)κ∗θ)ψ. For the mapping

Lµ;ℓ(cl)(R

n; Rq; CN− ,CN+) ∋ A(λ) 7→ κ∗θA(λ)κ∗θ ∈ Lµ;ℓcomp (cl)(U ; Rq; CN− ,CN+)

is continuous, we obtain b) from a). c) follows immediately from a) and b), whiled) is a consequence of c).


Remark 3.1.4. The mapping (3.1.2) gives rise to a continuous mapping opxN×j=1

Sµ;ℓ(cl)(R

n × Rn × Rq; CN− ,CN+) × L−∞(X ; Rq;E,F ) ∋ (a1, . . . , aN ;K(λ))

7−→N∑

j=1

ϕjκ∗j (opx(aj)(λ))ψj +K(λ) ∈ Lµ;ℓ

(cl)(X ; Rq;E,F ),

(3.1.3)

the so called operator convention for parameter-dependent pseudodifferential op-erators, while (3.1.1) induces a continuous mapping

Lµ;ℓ(cl)(X ; Rq;E,F ) ∋ A(λ) 7−→ ((a1, . . . , aN );K(λ))

∈N×j=1

Sµ;ℓ(cl)(R

n × Rn × Rq; CN− ,CN+) × L−∞(X ; Rq;E,F ),

(3.1.4)

which is right-inverse to (3.1.3).

We call a tuple (a1, . . . , aN ) ∈N×j=1

Sµ;ℓ(cl)(R

n × Rn ×Rq; CN− ,CN+) a complete

symbol for the operator A(λ) ∈ Lµ;ℓ(cl)(X ; Rq;E,F ), if the following conditions are

fulfilled:

• For any ϕ, ψ ∈ C∞0 (Ωj) it holds κj,∗(ϕA(λ)ψ) = (κj,∗ϕ)opx(aj)(λ)(κj,∗ψ)

modulo L−∞(Rn; Rq; CN− ,CN+).

• For any choice of the partition of unity ϕjj and functions ψjj in No-

tation 3.1.1 we have A(λ) −N∑j=1

ϕjκ∗j (opx(aj)(λ))ψj ∈ L−∞(X ; Rq;E,F ).

The tuple (a1, . . . , aN ) obtained from (3.1.4) yields a complete symbol for theoperator A(λ). We will refer to the mapping A(λ) 7→ (a1, . . . , aN ) also as thesymbol mapping.

In the classical case, equation (2.2.10) in Theorem 2.2.15 shows, that with

A(λ) ∈ Lµ;ℓcl (X ; Rq;E,F ) we can associate uniquely the principal symbol

σµ;ℓψ (A) ∈ C∞((T ∗X × Rq) \ 0,Hom(π∗E, π∗F )) (3.1.5)

which is anisotropic homogeneous of degree µ in the fibres of (T ∗X×Rq)\0. Here π∗

denotes the pull-back with respect to the projection π : (T ∗X ×Rq) \ 0 → X . The

mapping A(λ) 7→ σµ;ℓψ (A) is continuous. The following sequence is topologically

exact and splits:

0 −→Lµ−1;ℓcl (X ; Rq;E,F )

ı−→ Lµ;ℓcl (X ; Rq;E,F )

σµ;ℓψ−→

S(µ;ℓ)((T ∗X × Rq) \ 0,Hom(π∗E, π∗F )) −→ 0,(3.1.6)

where S(µ;ℓ)((T ∗X × Rq) \ 0,Hom(π∗E, π∗F )) denotes the space of anisotropichomogeneous functions of degree µ as a closed subspace of C∞((T ∗X × Rq) \0,Hom(π∗E, π∗F )).


Theorem 3.1.5. Let A(λ) ∈ Lµ;ℓ(X ; Rq;E,F ). Then A(λ) extends by continuityto a family of continuous operators A(λ) : Hs(X,E) → Hs−ν(X,F ) for everys, ν ∈ R with ν ≥ µ. The following estimates for the norms are valid:

‖A(λ)‖L(Hs(X,E),Hs−ν(X,F )) ≤Cs,ν〈λ〉

µℓ ν ≥ 0

Cs,ν〈λ〉µ−νℓ ν ≤ 0,

where Cs,ν > 0 are suitable constants depending on s, ν and A(λ), which may bechosen uniformly for A(λ) in bounded subsets of Lµ;ℓ(X ; Rq;E,F ). More precisely,this induces a continuous embedding

Lµ;ℓ(X ; Rq;E,F ) →Sµℓ (Rq;Hs(X,E), Hs−ν(X,F )) ν ≥ 0

Sµ−νℓ (Rq;Hs(X,E), Hs−ν(X,F )) ν ≤ 0


Proof. This follows by means of Theorem 3.1.3 from Theorem 2.2.13.

Theorem 3.1.6. a) Let G be another vector bundle over X. Then the compositionof operators on C∞(X,E) gives rise to a continuous bilinear mapping

Lµ;ℓ(cl)(X ; Rq;G,F ) × Lµ

′;ℓ(cl) (X ; Rq;E,G) → Lµ+µ′;ℓ

(cl) (X ; Rq;E,F )

for µ, µ′ ∈ R. If (a1, . . . , aN ) is a complete symbol for the operator A(λ) ∈Lµ;ℓ

(cl)(X ; Rq;G,F ), and (b1, . . . , bN ) a complete symbol for the operator B(λ) ∈Lµ

′;ℓ(cl) (X ; Rq;E,G), then (a1#b1, . . . , aN#bN ) is a complete symbol for the com-

position A(λ)B(λ) ∈ Lµ+µ′;ℓ(cl) (X ; Rq;E,F ).

In particular, the operators of order −∞ remain invariant with respect tocompositions from the left and from the right, i. e., they share the properties ofa two-sided ideal in the pseudodifferential operators.

If A(λ) and B(λ) are classical, then the following relation holds for the prin-cipal symbol of the composition:

σµ+µ′;ℓψ (AB) = σµ;ℓ

ψ (A) · σµ′;ℓψ (B).

b) Let A(λ) ∈ Lµ;ℓ(cl)(X ; Rq;E,F ). Then the formal adjoint operator belongs to

Lµ;ℓ(cl)(X ; Rq;F,E). Moreover, this induces a continuous antilinear mapping

Lµ;ℓ(cl)(X ; Rq;E,F ) ∋ A(λ) 7→ A(λ)(∗) ∈ Lµ;ℓ

(cl)(X ; Rq;F,E).

In the classical case, we have the following relation for the homogeneous

principal symbol: σµ;ℓψ (A(∗)) = σµ;ℓ

ψ (A)∗.

Proof. In the proof we suppress the vector bundles from the notation for betterreadability.

To prove a), note that by Theorem 3.1.5 and the closed graph theorem we justhave to show that the spaces of (classical) parameter-dependent pseudodifferentialoperators remain invariant under composition, and secondly that the formulae forthe symbols are valid.


According to Definition 3.1.2 and Theorem 3.1.5 the space L−∞(X ; Rq)clearly remains invariant with respect to compositions from the left and fromthe right with operators in Lµ;ℓ(X ; Rq), i. e.,

Lµ;ℓ(X ; Rq) × L−∞(X ; Rq) → L−∞(X ; Rq)

L−∞(X ; Rq) × Lµ;ℓ(X ; Rq) → L−∞(X ; Rq).

Let ϕ, ψ ∈ C∞(X) such that supp(ϕ)∩supp(ψ) = ∅. Choose a function η ∈ C∞(X)with supp(η) ∩ supp(ϕ) = ∅, and η ≡ 1 in a neighbourhood of supp(ψ). Hence weobtain

ϕA(λ)B(λ)ψ =(ϕA(λ)η

)︸︷︷︸∈L−∞(X;Rq)

B(λ)ψ + ϕA(λ)((1 − η)B(λ)ψ

)︸︷︷︸

∈L−∞(X;Rq)

∈ L−∞(X ; Rq)

as desired.Now let ϕ, ψ ∈ C∞

0 (Ω) be supported in the same coordinate neighbourhood(κ,Ω, U). Choose a function θ ∈ C∞

0 (Ω) such that θ ≡ 1 in a neighbourhood ofthe support of ϕ and ψ. Then we see

ϕA(λ)B(λ)ψ = ϕA(λ)θ2B(λ)ψ + ϕA(λ)((1 − θ2)B(λ)ψ

)︸︷︷︸

∈L−∞(X;Rq)

= (ϕA(λ)θ)(θB(λ)ψ) +R(λ)

with R(λ) ∈ L−∞(X ; Rq). For

κ∗((ϕA(λ)θ)(θB(λ)ψ)) = κ∗(ϕA(λ)θ) · κ∗(θB(λ)ψ) ∈ Lµ+µ′;ℓ(cl) (Rn; Rq)

according to Definition 3.1.2 and Theorem 2.2.4, we finally obtain the desiredassertion about the composition. Moreover, the latter identity also implies thecorresponding results about the complete symbol and the homogeneous principalsymbol.

To show b) note first that the assertion is immediately clear for operatorsbelonging to L−∞(X ; Rq). Consequently, we may restrict ourselves to operatorsthat are supported in a coordinate neighbourhood (κ,Ω, U), such that the bundlesare trivial over Ω. But in this case we may apply locally Theorem 2.2.4 from whichwe deduce the desired result.

Lemma 3.1.7. There exists a family of operators Hθ; θ ∈ R+ on the space⋃µ∈R

Lµ;ℓ(X ; Rq;E,F ) of parameter-dependent pseudodifferential operators with the

following properties:

Hθ : Lµ;ℓ(cl)(X ; Rq;E,F ) −→ Lµ;ℓ

(cl)(X ; Rq;E,F ),

I −Hθ : Lµ;ℓ(cl)(X ; Rq;E,F ) −→ L−∞(X ; Rq;E,F )

are continuous for each µ ∈ R. Moreover, given a sequence (µk) ⊆ R suchthat µk > µk+1 −→

k→∞−∞ and countable systems of bounded sets (Akj )j∈N ⊆


Lµk;ℓ(X ; Rq;E,F ), we may find a sequence (ci) ⊆ R+ with ci < ci+1 −→k→∞

∞having the property, that for each k ∈ N0

∞∑

i=k

supa∈Aij

p(Hdia) <∞

for all continuous seminorms p on Lµk;ℓ(X ; Rq;E,F ) and every j ∈ N, for allsequences (di) ⊆ R+ with di ≥ ci. If the sequence (µk)k∈N0 is given as µk := µ− kand the subsets are bounded in the classical operators, the same assertion holds for

all continuous seminorms p on Lµk;ℓcl (X ; Rq;E,F ).

Proof. In view of the operator convention and the symbol mappings it suffices toconstruct the operators Hθ on the symbol spaces and on L−∞(X ; Rq;E,F ) withthe corresponding properties there (cf. Remark 3.1.4). Let χ1 ∈ C∞(Rn×Rq) andχ2 ∈ C∞(Rq) be 0-excision functions. Define for θ ∈ R+ the operators Hθ onSµ;ℓ(Rn × Rn × Rq; CN− ,CN+) and on L−∞(X ; Rq;E,F ), respectively, via

(Hθa)(x, ξ, λ) := χ1

(ξθ,λ

θℓ)· a(x, ξ, λ) for a ∈ Sµ;ℓ(Rn × Rn × Rq; CN− ,CN+),

(HθA)(λ) := χ2

(λθ

)·A(λ) for A(λ) ∈ L−∞(X ; Rq;E,F ).

Then we obtain the assertion from Lemma 2.1.7 (see also Remark 2.1.10).


−∞and µ := max

k∈N

µk. Moreover, let Ak(λ) ∈ Lµk;ℓ(X ; Rq;E,F ). An operator A(λ) ∈Lµ;ℓ(X ; Rq;E,F ) is called the asymptotic expansion of the Ak(λ), if for everyR ∈ R there is a k0 ∈ N such that for k > k0

A(λ) −k∑

j=1

Aj(λ) ∈ LR;ℓ(X ; Rq;E,F ).

The operator A(λ) is uniquely determined up to L−∞(X ; Rq;E,F ).In analogous manner as we proved Theorem 2.1.8 using Lemma 2.1.7, we now

obtain from Lemma 3.1.7 corresponding existence results of operators (or operatorfamilies) having a prescribed asymptotic expansion.

Ellipticity and parametrices.

Definition 3.1.9. Let A(λ) ∈ Lµ;ℓ(cl)(X ; Rq;E,F ). Then A(λ) is called parameter-

dependent elliptic, if the following condition is fulfilled:For every compact set K ⋐ (κ,Ω, U) contained in a coordinate neighbour-

hood (with the vector bundles being trivial over Ω), and every ϕ, ψ ∈ C∞0 (Ω) such

that ϕ, ψ ≡ 1 on K, the push-forward κ∗(ϕA(λ)ψ) ∈ Lµ;ℓcomp (cl)(U ; Rq; CN− ,CN+)

is parameter-dependent elliptic on κ(K) ⋐ U in the sense of Definition 2.2.8.


In the classical case, the condition of parameter-dependent ellipticity simpli-

fies to the invertibility of σµ;ℓψ (A)(x, ξ, λ) for (x, ξ, λ) ∈

(T ∗X×Rq

)\0, see Remark

2.2.9.Note that for the existence of parameter-dependent elliptic elements it is

necessary that the dimensions of the vector bundles coincide, i. e., N− = N+.

Theorem 3.1.10. Let A(λ) ∈ Lµ;ℓ(X ; Rq;E,F ). The following assertions areequivalent:

a) A(λ) is parameter-dependent elliptic.b) The components of the complete symbol (a1, . . . , aN ) of A(λ) are parameter-

dependent elliptic on κj(suppψj) for j = 1, . . . , N .c) There exists an operator P (λ) ∈ L−µ;ℓ(X ; Rq;F,E) such that A(λ)P (λ) − I ∈

L−ε;ℓ(X ; Rq;F, F ) and P (λ)A(λ) − I ∈ L−ε;ℓ(X ; Rq;E,E) for some ε > 0.d) There exists an operator P (λ) ∈ L−µ;ℓ(X ; Rq;F,E) such that A(λ)P (λ) − I ∈

L−∞(X ; Rq;F, F ) and P (λ)A(λ) − I ∈ L−∞(X ; Rq;E,E).

Moreover, if A(λ) ∈ Lµ;ℓcl (X ; Rq;E,F ) is parameter-dependent elliptic, then every

operator P (λ) satisfying d) belongs to L−µ;ℓcl (X ; Rq;F,E). Every P (λ) satisfying

d) is called a (parameter-dependent) parametrix of A(λ).

Proof. a) implies b) follows from the definition of parameter-dependent ellip-ticity. Now assume that b) holds. From Corollary 2.2.11 we obtain the exis-

tence of b1, . . . , bN ∈ S−µ;ℓ(cl) (Rn × Rn × Rq; CN+ ,CN−) and suitable functions

ϕj , ψj ∈ C∞0 (Rn) such that ψjϕj = ϕj and ϕj ≡ 1 on κj(suppψj) ⋐ Rn with the

property that ϕj(opx(aj)(λ)opx(bj)(λ)−1)ψj belongs to L−∞(Rn; Rq; CN+ ,CN+).Now define P (λ) := opx((b1, . . . , bN); 0) with the operator convention (3.1.3),

which yields an operator P (λ) ∈ L−µ;ℓ(cl) (X ; Rq;F,E). Now it is straightforward

to check that A(λ)P (λ) − I belongs to L−1;ℓ(cl) (X ; Rq;F, F ). Analogously, we ob-

tain a parametrix from the left. But both the left- and the right-parametrix differonly by a term of order −1 which gives c). c) implies d) follows by means of aformal Neumann series argument as, e. g., in the proof of Theorem 2.2.10, wherenow Theorem 3.1.6 and Definition 3.1.8 enter the argument. This also yields theexistence of a classical parametrix if we started with an elliptic classical parameter-dependent operator A(λ). d) implies a) is part of Corollary 2.2.11 when passingto local coordinates.

Theorem 3.1.11. Let ∆ be a compact C∞-manifold (not necessarily with empty

boundary), and ∆ ∋ δ 7−→ Aδ(λ) ∈ Lµ;ℓ(cl)(X ; Rq;E,F ) be a smooth family that is

locally uniformly parameter-dependent elliptic. Then the set

K := (δ, λ) ∈ ∆ × Rq;(Aδ(λ)

)−1 ∈ L(Hs−µ(X,F ), Hs(X,E)) does not existis compact in ∆ × Rq and independent of s ∈ R. Moreover, for any given neigh-bourhood U(K) ⊆ ∆ × Rq of K, there exists a C∞-family ∆ ∋ δ 7→ P δ(λ) ∈L−µ;ℓ

(cl) (X ; Rq;F,E) such that Aδ(λ)P δ(λ)− I and P δ(λ)Aδ(λ)− I depend smoothly


on δ ∈ ∆ with values in the operators of order −∞, and P δ(λ) = (Aδ(λ))−1 for(δ, λ) ∈

(∆ × Rq

)\ U(K).

Proof. In the proof we omit the vector bundles for better readability. From ellipticregularity we obtain that the set K is indeed independent of s ∈ R. Consequently,

we may fix s ∈ R in the sequel. Let Bδ(λ) ∈ L−µ;ℓ(cl) (X ; Rq) be C∞ such that

Aδ(λ)Bδ(λ) − I =: RδR(λ) and Bδ(λ)Aδ(λ) − I =: RδL(λ) depend smoothly onδ ∈ ∆ with values in L−∞(X ; Rq) (cf. Theorem 3.1.10). From Theorem 3.1.5 (orthe defining characterization of L−∞(X ; Rq) in Definition 3.1.2) we see, that for|λ| sufficiently large and all δ ∈ ∆ the operators I + RδR(λ) and I + RδL(λ) areinvertible in L(Hs−µ(X)) and L(Hs(X)), respectively. Thus it remains to showthe closedness of the set K ⊆

(∆ × Rq

). For the set of invertible operators in

L(Hs(X), Hs−µ(X)) is open, and since Aδ(λ) may be viewed as a continuousfunction of (δ, λ) ∈ ∆×Rq with values in this space, we obtain the closedness andconsequently the asserted compactness of K ⊆ ∆ × Rq.

Now let U(K) ⊆ ∆×Rq be any given neighbourhood of K. Let χ ∈ C∞(∆×Rq) such that χ ≡ 0 on K and χ ≡ 1 on

(∆ × Rq

)\ U(K). Define

P δL(λ) := Bδ(λ) −RδL(λ)Bδ(λ) +RδL(λ)χ(δ, λ)(Aδ(λ))−1RδR(λ),

P δR(λ) := Bδ(λ) −Bδ(λ)RδR(λ) +RδL(λ)χ(δ, λ)(Aδ(λ))−1RδR(λ).

From Theorem 3.1.5 we obtain that RδL(λ)χ(δ, λ)(Aδ(λ))−1RδR(λ) is a C∞-functionof δ ∈ ∆ taking values in L−∞(X ; Rq), and thus P δL(λ) and P δR(λ) depend smoothly

on δ ∈ ∆ with values in L−µ;ℓ(cl) (X ; Rq). Moreover, we have P δL(λ)Aδ(λ) = I as well

as Aδ(λ)P δR(λ) = I for (δ, λ) ∈(∆×Rq

)\U(K). Now define P δ(λ) either as P δL(λ)

or as P δR(λ), which concludes the proof.

Theorem 3.1.12. There exist parameter-dependent reductions of orders, i. e.,there exist operators Rµ(λ) ∈ Lµ;ℓ(X ; Rq;E,E) such that Rµ(λ)R−µ(λ) = I forevery µ ∈ R.

Proof. For µ = 0 choose the identity. Now let µ > 0. With the given Riemannianmetric on X we define the anisotropic homogeneous function of degree µ on T ∗X×Rq+1 \ 0 with values in Hom(π∗E) via

a(µ)(ξx, (λ, λq+1)) :=(〈ξx, ξx〉ℓx + |(λ, λq+1)|2

) µ2ℓ · idπ∗E(ξx,(λ,λq+1))

.

The associated operator in Lµ;ℓ(X ; Rq+1;E,E) therefore is parameter-dependentelliptic. From Theorem 3.1.11 we now obtain, that if we fix λq+1 with |λq+1|sufficiently large, we obtain invertible operators Rµ(λ) ∈ Lµ;ℓ(X ; Rq;E,E) withinverses R−µ(λ) in L−µ;ℓ(X ; Rq;E,E). This proves the theorem.

Remark 3.1.13. In analogous manner, we also have the calculus of anisotropicparameter-dependent pseudodifferential operators on closed manifolds X , wherethe parameter-space Rq is substituted by a conical set Λ ⊆ Rq that is assumed to


be the closure of its interior. This will be employed, in particular, with half-planesin C ∼= R2.

3.2. Parameter-dependent Volterra operators

Remark 3.2.1. Throughout this section we employ again the notations from No-tation 3.1.1 with the same data fixed on X and the vector bundles E and F .

Definition 3.2.2. a) Define

L−∞V (X ; H;E,F ) := L−∞(X ; H;E,F ) ∩ A(

H, L−∞(X ;E,F )),

which is a closed subspace of L−∞(X ; H;E,F ). It is characterized as

S(H,⋂

s,t∈R

L(Hs(X,E), Ht(X,F ))) ∩A(

H,⋂

s,t∈R

L(Hs(X,E), Ht(X,F )))

=⋂

s,t∈R

S(H,L(Hs(X,E), Ht(X,F ))) ∩ A(

H,L(Hs(X,E), Ht(X,F )))

= S−∞V (H; ind-lim

s∈R

Hs(X,E), proj-limt∈R

Ht(X,F )).

b) For µ ∈ R define

Lµ;ℓV (cl)(X ; H;E,F ) := Lµ;ℓ

(cl)(X ; H;E,F ) ∩ A(

H, Lµ(X ;E,F )),

with the space of pseudodifferential operators Lµ(X ;E,F ) on X acting in sec-

tions of the vector bundles E and F . Thus Lµ;ℓV (cl)(X ; H;E,F ) becomes a closed

subspace of Lµ;ℓ(cl)(X ; H;E,F ).

Remark 3.2.3. From the considerations about the calculus of Volterra pseudodif-ferential operators in Section 2.4 on the one hand as well as Definition 3.1.2 on the

other hand we conclude, that the space Lµ;ℓV (cl)(X ; H;E,F ) consists of all families

of operators A(λ) : C∞(X,E) → C∞(X,F ) with the following properties:

i) For all functions ϕ, ψ ∈ C∞(X) that are supported in the same coor-dinate neighbourhood (κ,Ω, U) the push-forward κ∗(ϕA(λ)ψ) belongs to

Lµ;ℓcompV (cl)(U ; H; CN− ,CN+).

ii) For all ϕ, ψ ∈ C∞(X) having disjoint support the operator ϕA(λ)ψ is anelement of L−∞

V (X ; H;E,F ).

The projective topology on the space Lµ;ℓV (cl)(X ; H;E,F ) with respect to the map-

pings A(λ) 7−→κ∗(ϕA(λ)ψ) ∈ Lµ;ℓ

compV (cl)(U ; H; CN− ,CN+) for ϕ, ψ supported in (κ,Ω, U)

ϕA(λ)ψ ∈ L−∞V (X ; H;E,F ) for suppϕ ∩ suppψ = ∅

is exactly the given one from Definition 3.2.2.Theorem 3.1.3 holds within parameter-dependent Volterra pseudodifferential

operators:


a) Let (κ,Ω, U) be a chart and A(λ) ∈ Lµ;ℓcompV (cl)(U ; H; CN− ,CN+). Then the

pull-back κ∗(A(λ)) belongs to Lµ;ℓV (cl)(X ; H;E,F ) and provides a continuous

linear mapping

Lµ;ℓcompV (cl)(U ; H; CN− ,CN+) ∋ A(λ) 7−→ κ∗(A(λ)) ∈ Lµ;ℓ

V (cl)(X ; H;E,F ).

b) Let (κ,Ω, U) be a chart and ϕ, ψ ∈ C∞0 (Ω).

Moreover, let A(λ) ∈ Lµ;ℓV (cl)(R

n; H; CN− ,CN+). Then the operator ϕκ∗(A(λ))ψ

belongs to Lµ;ℓV (cl)(X ; H;E,F ), and the mapping

Lµ;ℓV (cl)(R

n; H; CN− ,CN+) ∋ A(λ) 7−→ ϕκ∗(A(λ))ψ ∈ Lµ;ℓV (cl)(X ; H;E,F )

is continuous.c) The restriction of the mapping (3.1.1)

Lµ;ℓV (cl)(X ; H;E,F ) −→

N×j=1

Lµ;ℓcompV (cl)(Uj ; H; CN− ,CN+) × L−∞

V (X ; H;E,F )

(3.2.1)to Volterra pseudodifferential operators is well-defined and continuous, and sois the restriction of the mapping (3.1.2)

N×j=1

Lµ;ℓV (cl)(R

n; H; CN− ,CN+) × L−∞V (X ; H;E,F ) −→ Lµ;ℓ

V (cl)(X ; H;E,F ). (3.2.2)

d) For β ∈ N0 the complex derivative acts continuously in the spaces ∂βλ :

Lµ;ℓV (cl)(X ; H;E,F ) → L

µ−ℓ|β|;ℓV (cl) (X ; H;E,F ).

The restriction of the parameter to the real line induces a continuous embedding

Lµ;ℓV (cl)(X ; H;E,F ) → Lµ;ℓ

(cl)(X ; R;E,F ).

This follows from Proposition 2.3.2, Remark 2.4.4.

Remark 3.2.4. From (3.2.2) and (3.2.1) we see that the restriction of the operatorconvention opx from (3.1.3)

N×j=1

Sµ;ℓV (cl)(R

n × Rn × H; CN− ,CN+) × L−∞V (X ; H;E,F ) −→ Lµ;ℓ

V (cl)(X ; H;E,F ),

(3.2.3)as well as the restriction of the mapping (3.1.4)


N×j=1

Sµ;ℓV (cl)(R

n × Rn × H; CN− ,CN+) × L−∞V (X ; H;E,F )

(3.2.4)to Volterra symbols respectively operators are well-defined and continuous.The symbol mapping induced from (3.2.4) now associates to a given operator

A(λ) ∈ Lµ;ℓV (cl)(X ; H;E,F ) a complete symbol (a1, . . . , aN ) consisting of (classical)

Volterra symbols.In the classical case, the parameter-dependent anisotropic homogeneous prin-

cipal symbol σµ;ℓψ (A) ∈ C∞((T ∗X × H) \ 0,Hom(π∗E, π∗F )) of an operator


A(λ) ∈ Lµ;ℓV cl(X ; H;E,F ) is analytic in the interior of H. This follows from Propo-

sition 2.3.2.

Theorem 3.2.5. Let G be another vector bundle over X. Then the compositionof operators (cf. Theorem 3.1.6) restricts to a continuous bilinear mapping

Lµ;ℓV (cl)(X ; H;G,F ) × Lµ

′;ℓV (cl)(X ; H;E,G) → Lµ+µ′;ℓ

V (cl) (X ; H;E,F )

for µ, µ′ ∈ R. In particular, the Volterra operators of order −∞ remain invariantwith respect to compositions from the left and from the right, i. e., they share theproperties of a two-sided ideal in the Volterra pseudodifferential operators.

Proof. By Theorem 3.1.6 the bilinear mappings

A(

H, Lµ(X ;G,F )) ×A(

H, Lµ′

(X ;E,G)) −→ A(

H, Lµ+µ′

(X ;E,F )),

Lµ;ℓ(cl)(X ; H;G,F ) × Lµ

′;ℓ(cl) (X ; H;E,G) −→ Lµ+µ′;ℓ

(cl) (X ; H;E,F )

are well-defined and continuous. This implies the assertion in view of Definition3.2.2.

Theorem 3.2.6. Let A(λ) ∈ Lµ;ℓV (X ; H;E,F ). Then A(λ) extends by continuity

to a family of continuous operators A(λ) : Hs(X,E) → Hs−ν(X,F ) for everys, ν ∈ R with ν ≥ µ. Moreover, this induces a continuous embedding

Lµ;ℓV (X ; H;E,F ) →

Sµℓ

V (H;Hs(X,E), Hs−ν(X,F )) ν ≥ 0

Sµ−νℓ

V (H;Hs(X,E), Hs−ν(X,F )) ν ≤ 0

into the space of operator-valued Volterra symbols in the Sobolev spaces.

Proof. This follows from Theorem 3.1.5 and Definition 3.2.2.

Kernel cut-off behaviour and asymptotic expansion.


−∞and µ := max

k∈N

µk. Moreover, let Ak(λ) ∈ Lµk;ℓV (X ; H;E,F ). An operator A(λ) ∈Lµ;ℓV (X ; H;E,F ) is called the asymptotic expansion of the Ak(λ), if for every R ∈ R

there is a k0 ∈ N such that for k > k0

A(λ) −k∑

j=1

Aj(λ) ∈ LR;ℓV (X ; H;E,F ).

The operator A(λ) is uniquely determined up to L−∞V (X ; H;E,F ). For short, we

write A(λ) ∼V

∞∑j=1

Aj(λ).

Note that the distinction between the notion of asymptotic expansion fromDefinition 3.1.8 is that we require the remainders to have the Volterra property(cf. Definition 2.3.3). In order to obtain existence results of Volterra operatorshaving a prescribed asymptotic expansion we need to carry over the considerations


concerning the kernel cut-off operator (cf. Definition 2.3.5, Theorem 2.3.6, andCorollary 2.3.7) and deduce an analogue of Proposition 2.3.8.

Remark 3.2.8. Let µ ∈ R and µ+ := max0, µ. From Theorem 3.1.5 and Theorem3.2.6 we see that for every s ∈ R we have continuous embeddings

Lµ;ℓ(X ; R;E,F ) → Sµℓ (R;Hs(X,E), Hs−µ+(X,F )),

Lµ;ℓV (X ; H;E,F ) → S

µℓ

V (H;Hs(X,E), Hs−µ+(X,F )).

By Theorem 2.3.6 the kernel cut-off operator H (see Definition 2.3.5) acts as abilinear and continuous map in the spaces

C∞b (R)×S µ

ℓ (R;Hs(X,E), Hs−µ+(X,F )) −→ Sµℓ (R;Hs(X,E), Hs−µ+(X,F )),

C∞b (R)×S

µℓ

V (H;Hs(X,E), Hs−µ+(X,F )) −→ Sµℓ

V (H;Hs(X,E), Hs−µ+(X,F )).

Theorem 3.2.9. The kernel cut-off operator H (cf. Remark 3.2.8) restricts tocontinuous bilinear mappings

H :

C∞b (R)×Lµ;ℓ

(cl)(X ; R;E,F ) −→ Lµ;ℓ(cl)(X ; R;E,F )

C∞b (R)×Lµ;ℓ

V (cl)(X ; H;E,F ) −→ Lµ;ℓV (cl)(X ; H;E,F ).

Moreover, the following asymptotic expansion (in the sense of Definitions 3.1.8,3.2.7) holds for

(H(ϕ)A

)(λ) in terms of ϕ and A(λ):

(H(ϕ)A

)(λ) ∼

(V )

∞∑

k=0

((−1)k

k!Dkt ϕ(0)

)·(∂kλA

)(λ)

where ∂λ denotes the complex derivative with respect to λ ∈ H in case of Volterraoperators.

Proof. For the proof of the first assertion we simply have to check that H mapsthe corresponding spaces into each other as asserted. The (separate) continuitythen follows from the closed graph theorem.

Employing (3.1.4) and (3.2.4) we obtain continuous linear mappings

Lµ;ℓ(cl)(X ; R;E,F ) −→

N×j=1

Sµ;ℓ(cl)(R

n × Rn × R; CN− ,CN+) × L−∞(X ; R;E,F ),


N×j=1

Sµ;ℓV (cl)(R

n × Rn × H; CN− ,CN+) × L−∞V (X ; H;E,F ).

From Theorem 2.3.6 we conclude that the kernel cut-off operator acts bilinear andcontinuous in each of the factors on the right-hand sides, i. e.,

C∞b (R)×Sµ;ℓ

(cl)(Rn × Rn × R; CN− ,CN+) −→ Sµ;ℓ

(cl)(Rn × Rn × R; CN− ,CN+)

C∞b (R)×Sµ;ℓ

V (cl)(Rn × Rn × H; CN− ,CN+) −→ Sµ;ℓ

V (cl)(Rn × Rn × H; CN− ,CN+),


as well as

C∞b (R)×L−∞(X ; R;E,F ) −→ L−∞(X ; R;E,F )

C∞b (R)×L−∞

V (X ; H;E,F ) −→ L−∞V (X ; H;E,F ),

keeping in mind the characterizations of L−∞(X ; R;E,F ) and L−∞V (X ; H;E,F )

as operator-valued symbols from Definition 3.1.2 and Definition 3.2.2. More-over, the asymptotic expansion (2.3.2) holds in the factors corresponding to theL(CN− ,CN+)-valued symbols.

Now we see that we find the kernel cut-off operator (restricted to parameter-dependent pseudodifferential operators as in the assertion of the theorem) as com-position of the mappings (3.1.4) resp. (3.2.4), the “local” kernel cut-off operatorsin the factors as discussed above, and the operator convention (3.1.3) resp. (3.2.3).Moreover, the “local” expansions give also the second assertion of the theoremconcerning the asymptotic expansion.

Corollary 3.2.10. Let ϕ ∈ C∞0 (R) such that ϕ ≡ 1 near t = 0. Then the operator

I −H(ϕ) acts continuous in the spaces

I −H(ϕ) :

Lµ;ℓ(X ; R;E,F ) −→ L−∞(X ; R;E,F )

Lµ;ℓV (X ; H;E,F ) −→ L−∞

V (X ; H;E,F ).

Proposition 3.2.11. Let (µk) ⊆ R such that µk ≥ µk+1 −→k→∞

−∞. Furthermore,

for each k ∈ N let (Akj )j∈N ⊆ Lµk;ℓV (X ; H;E,F ) be a countable system of boundedsets. Let ϕ ∈ C∞

0 (R), and for c ∈ [1,∞) let ϕc ∈ C∞0 (R) be defined as ϕc(t) :=

ϕ(ct). Then there is a sequence (ci) ⊆ [1,∞) with ci < ci+1 −→i→∞

∞ such that for

each k ∈ N∞∑

i=k

supA(λ)∈Aij

p((H(ϕdi)A)(λ)

)<∞

for all continuous seminorms p on Lµk;ℓV (X ; H;E,F ) and every j ∈ N, and for allsequences (di) ⊆ R+ with di ≥ ci.

Proof. Employing (3.2.4) and the operator convention (3.2.3) as well as Theorem

3.2.9 reduces the proof to the case of Volterra symbol spaces Sµk;ℓV (Rn × Rn ×H; CN− ,CN+) and L−∞

V (X ; H;E,F ). But for these the assertion follows at oncefrom Proposition 2.3.8.

Theorem 3.2.12. Let (µk) ⊆ R such that µk −→k→∞

−∞ and µ := maxk∈N0

µk. More-

over, let Ak(λ) ∈ Lµk;ℓV (X ; H;E,F ). Then there exists A(λ) ∈ Lµ;ℓV (X ; H;E,F )

such that A(λ) ∼V

∞∑k=0

Ak(λ). The asymptotic sum A(λ) is uniquely determined

modulo L−∞V (X ; H;E,F ).

If the sequence (µk)k∈N0 is given as µk = µ − k with classical operators

Ak(λ) ∈ Lµ−k;ℓV cl (X ; H;E,F ), then also A(λ) ∈ Lµ;ℓV cl(X ; H;E,F ).


Proof. This follows analogously to the proof of Theorem 2.3.9, but now Proposition3.2.11 enters the argument replacing Proposition 2.3.8 which was used there.

The translation operator in Volterra pseudodifferential operators.

Remark 3.2.13. Let µ ∈ R and µ+ := max0, µ. According to Proposition 2.3.11the translation operator Tiτ for τ ≥ 0 (cf. Definition 2.3.10) acts as a linearcontinuous operator in the spaces

Tiτ : Sµℓ

V (H;Hs(X,E), Hs−µ+(X,F )) −→ Sµℓ

V (H;Hs(X,E), Hs−µ+(X,F ))

for every s ∈ R.

Proposition 3.2.14. For every τ ≥ 0 the translation operator Tiτ (cf. Remarks3.2.13, 3.2.8) restricts to a linear and continuous operator in the spaces

Tiτ : Lµ;ℓV (cl)(X ; H;E,F ) −→ Lµ;ℓ

V (cl)(X ; H;E,F ).

Moreover,(TiτA

)(λ) has the following asymptotic expansion in terms of τ and

A(λ) (in the sense of Definition 3.2.7):

(TiτA

)(λ) ∼

V

∞∑

k=0

(iτ)k

k!·(∂kλA

)(λ).

In particular, the operator I − Tiτ is continuous in the spaces

I − Tiτ : Lµ;ℓV (cl)(X ; H;E,F ) −→ Lµ−ℓ;ℓV (cl) (X ; H;E,F ).

Proof. This follows from Proposition 2.3.11 when passing via (3.2.4) to “local”symbols and remainders in L−∞

V (X ; H;E,F ).

Notation 3.2.15. For every µ ∈ R let S(µ;ℓ)V ((T ∗X×H)\0,Hom(π∗E, π∗F )) denote

the space of anisotropic homogeneous functions of degree µ that are analytic inthe interior of H. This is a closed subspace of C∞((T ∗X×H)\0,Hom(π∗E, π∗F )).

Theorem 3.2.16. The restriction of the principal symbol sequence to Volterraoperators is topologically exact and splits:

0 −→Lµ−1;ℓV cl (X ; H;E,F )

ı−→ Lµ;ℓV cl(X ; H;E,F )

σµ;ℓψ−→

S(µ;ℓ)V ((T ∗X × H) \ 0,Hom(π∗E, π∗F )) −→ 0.

Proof. Every element in S(µ;ℓ)V ((T ∗X×H)\0,Hom(π∗E, π∗F )) can be represented

by a vector of local representatives corresponding to the given covering of X bycoordinate neighbourhoods from Notation 3.1.1 (i. e., they satisfy the transitionconditions of the bundles involved over the intersections). For every j = 1, . . . , N ,the representative over Ωj may be viewed as a C∞-function on Ωj taking values

in the space S(µ;ℓ)V

((Rn × H

)\ 0; CN−,CN+

)(cf. Notation 2.3.12). Thus, if we

multiply this function by θj and pass from Ωj to Uj via κj for every j = 1, . . . , N ,we get an N -tuple of compactly supported smooth functions on Rn taking values


in S(µ;ℓ)V

((Rn×H

)\0; CN−,CN+

). To each of these we now apply the translation

operator Tiτ for some τ > 0. By Theorem 2.3.13 we so obtained an N -tuple ofclassical L(CN− ,CN+)-valued Volterra symbols of order µ. Now we associate to this

tuple an operator in Lµ;ℓV cl(X ; H;E,F ) via (3.2.3). Summing up, we constructed a

continuous linear right-inverse to the principal symbol mapping

σµ;ℓψ : Lµ;ℓ

V cl(X ; H;E,F ) → S(µ;ℓ)V ((T ∗X × H) \ 0,Hom(π∗E, π∗F ))

which shows the assertion of the theorem.

Parabolicity for Volterra operators on manifolds.

Definition 3.2.17. Let A(λ) ∈ Lµ;ℓV (cl)(X ; H;E,F ). Then A(λ) is called parabolic,

if A(λ) is parameter-dependent elliptic as an element of Lµ;ℓ(cl)(X ; H;E,F ).

Theorem 3.2.18. Let A(λ) ∈ Lµ;ℓV (X ; H;E,F ). The following assertions are

equivalent:

a) A(λ) is parabolic.b) The components of the complete symbol (a1, . . . , aN ) of A(λ) are parabolic on

κj(suppψj) for j = 1, . . . , N .

c) There exists an operator P (λ) ∈ L−µ;ℓV (X ; H;F,E) such that A(λ)P (λ) − I ∈

L−ε;ℓV (X ; H;F, F ) and P (λ)A(λ) − I ∈ L−ε;ℓ

V (X ; H;E,E) for some ε > 0.

d) There exists an operator P (λ) ∈ L−µ;ℓV (X ; H;F,E) such that A(λ)P (λ) − I ∈

L−∞V (X ; H;F, F ) and P (λ)A(λ) − I ∈ L−∞

V (X ; H;E,E).

Moreover, if A(λ) ∈ Lµ;ℓV cl(X ; H;E,F ) is parabolic, then every P (λ) satisfying

d) belongs to L−µ;ℓV cl (X ; H;F,E). Every P (λ) satisfying d) is called a (parameter-

dependent) Volterra parametrix of A(λ).

Proof. From Theorem 3.1.10 and (3.2.4) and the definition of parabolicity asparameter-dependent ellipticity we conclude that we only have to prove that b)implies c), and c) implies d).

Now assume that b) holds. From Corollary 2.4.14 we obtain the existence

of b1, . . . , bN ∈ S−µ;ℓV (cl)(R

n × Rn × H; CN+ ,CN−) and suitable functions ϕj , ψj ∈C∞

0 (Rn) such that ψjϕj = ϕj and ϕj ≡ 1 on κj(suppψj) ⋐ Rn with the property

that ϕj(opx(aj)(λ)opx(bj)(λ) − 1)ψj belongs to L−∞V (Rn; H; CN+ ,CN+). Now de-

fine P (λ) := opx((b1, . . . , bN); 0) with the operator convention (3.2.3), which yields

an operator P (λ) ∈ L−µ;ℓV (cl)(X ; H;F,E). Now we see that A(λ)P (λ) − I belongs to

L−1;ℓV (cl)(X ; H;F, F ). Analogously, we obtain a (rough) Volterra parametrix from the

left. But both the left- and the right-parametrix differ only by a term of order −1which gives c).

c) implies d) follows analogously to the proof of Theorem 2.2.10 by meansof a formal Neumann series argument, where Theorem 3.2.5 and Theorem 3.2.12enter the argument for carrying out the compositions and asymptotic expansions.


Theorem 3.2.19. Let A(λ) ∈ Lµ;ℓV (cl)(X ; H;E,F ) be parabolic. Then the set

K := λ ∈ H;(A(λ)

)−1 ∈ L(Hs−µ(X,F ), Hs(X,E)) does not existis compact in H and independent of s ∈ R. Moreover, let τ ≥ 0 such that sup

λ∈K|λ| <

τ . Then the operator (TiτA)(λ) ∈ Lµ;ℓV (cl)(X ; H;E,F ) is invertible with inverse

((TiτA)(λ)

)−1 ∈ L−µ;ℓV (cl)(X ; H;F,E).

Proof. The first assertion follows from Theorem 3.1.11. In view of Proposition

3.2.14 this also implies that the operator (TiτA)(λ) ∈ Lµ;ℓV (cl)(X ; H;E,F ) is in-

vertible with inverse((TiτA)(λ)

)−1 ∈ L−µ;ℓ(cl) (X ; H;F,E). In particular, the op-

erator family (TiτA)(λ) ∈ A(

H, Lµ(X ;E,F )) is invertible with((TiτA)(λ)

)−1 ∈C∞(

H, L−µ(X ;F,E)). From the resolvent identity we conclude that for λ0, λ1 ∈

H

with λ0 6= λ1 the difference quotient

((TiτA)(λ0)

)−1

−((TiτA)(λ1)

)−1

λ0−λ1is given as

−((TiτA)(λ0)

)−1 ·( (TiτA)(λ0) − (TiτA)(λ1)

λ0 − λ1

)·((TiτA)(λ1)

)−1,

which implies the analyticity of((TiτA)(λ)

)−1in the interior of H.

Chapter 4. Weighted Sobolev spaces

4.1. Anisotropic Sobolev spaces on the infinite cylinder

Remark 4.1.1. In this chapter we again employ the notations from Notation 3.1.1with the corresponding data fixed on X and the vector bundle E.

The material in this section is standard in the isotropic case, i. e., ℓ = 1and t = 0 (in the notation of Definition 4.1.2 below). There are many variants ofanisotropic Sobolev spaces discussed in the literature, e. g., [2], [27], [49]. Therefore,we restrict ourselves to give the basic definitions and results in that form as theyare needed in this work.

Definition 4.1.2. For s, t ∈ R define the Sobolev space H(s,t);ℓ(R×X,E) as the

space of all u ∈ ⋃s′∈R

S′(R, Hs′(X,E)) such that Fu is a regular distribution and

‖u‖H(s,t);ℓ(R×X,E) :=(∫

R

‖Rs(τ)(Fr→τu)(τ)‖2Ht(X,E) dτ

) 12

<∞. (4.1.1)

Here Rs(τ) ∈ Ls;ℓ(X ; R;E,E) is a parameter-dependent reduction of orders fromTheorem 3.1.12.

Remark 4.1.3. a) The space H(s,t);ℓ(R ×X,E) is well-defined, i. e., other choicesof the reduction of orders give rise to equivalent norms (see Theorem 3.1.5, andTheorem 3.1.6).


b) For s = t = 0 we have H(0,0);ℓ(R ×X,E) = L2(R ×X,E) = L2(R, L2(X,E)),see also Proposition 4.1.7.

c) The local space H(s,0);ℓ(Rn+1,CN−) consists of all u ∈ S′(Rn+1,CN−) suchthat Fu is a regular distribution and

‖u‖H(s,0);ℓ(Rn+1,CN−) :=(∫

R

∫

Rn

〈ξ, τ〉2sℓ ‖(Fu)(ξ, τ)‖2CN−

dξ dτ) 1

2

<∞.

Using the covering of X by coordinate neighbourhoods as well as the sub-ordinated partition of unity from Notation 3.1.1 we see that the spaceH(s,0);ℓ(R ×X,E) consists precisely of those distributions that locally belongto H(s,0);ℓ(Rn+1,CN−).

Definition 4.1.4. For s, t, δ ∈ R define

H(s,t);ℓ(R ×X,E)δ := 〈r〉−δH(s,t);ℓ(R ×X,E) (4.1.2)

with the induced norm.

Theorem 4.1.5. a) H(s,t);ℓ(R×X,E)δ is a Hilbert space with respect to the scalarproduct

〈u, v〉 =

∫

R

〈Rs(τ)(Fr→τ 〈r〉δu(r)

)(τ), Rs(τ)

(Fr→τ 〈r〉δv(r)

)(τ)〉Ht(X,E) dτ.

b) The embedding S(R ×X,E) = S(R, C∞(X,E)) → H(s,t);ℓ(R ×X,E)δ is con-tinuous and dense.

c) The operator of multiplication with a function ϕ ∈ C∞b (R) induces a continuous

operator in L(H(s,t);ℓ(R × X,E)δ), and the mapping C∞b (R) ∋ ϕ 7→ Mϕ ∈

L(H(s,t);ℓ(R ×X,E)δ) is continuous.d) For s′ − s ≤ min0, t − t′ and δ ≥ δ′ the embedding H(s,t);ℓ(R × X,E)δ →

H(s′,t′);ℓ(R×X,E)δ′ is well-defined and continuous. Moreover, it is compact ifs′−s < min0, t−t′ and δ > δ′; Hilbert-Schmidt if s′−s+ n+ℓ

2 < min0, t−t′and δ − δ′ > 1

2 .

e) For k ∈ N0 let Ck;ℓb (R × X,E) denote the Banach space of all sectionsu : R ×X → E such that for |α|ℓ ≤ k there exists ∂α(x,t)u as a bounded contin-

uous function with respect to any choice of local coordinates and trivializationsof the vector bundle, endowed with the topology of uniform convergence of allderivatives up to (anisotropic) order k.

Sobolev embedding theorem: Let k ∈ N0. Then for s > k+ n+ℓ2 the embedding

H(s,0);ℓ(R ×X,E)δ → 〈t〉−δCk;ℓb (R ×X,E) is well-defined and continuous.

In particular, we have S(R ×X,E) =⋂

s,δ∈R

H(s,0);ℓ(R ×X,E)δ, which holds

topologically with the projective limit topology on the right-hand side.f) For every δ0 ∈ R the 〈r〉−δ0L2(R × X,E)-inner product extends to a non-

degenerate sesquilinear pairing

〈·, ·〉δ0 : H(s,t);ℓ(R ×X,E)δ0+δ ×H(−s,−t);ℓ(R ×X,E)δ0−δ −→ C


which induces an identification of the dual

H(s,t);ℓ(R ×X,E)′δ0+δ∼= H(−s,−t);ℓ(R ×X,E)δ0−δ.

In particular, this provides a topological (antilinear) isomorphism S(R ×X,E)′ ∼=

⋃s,δ∈R

H(s,0);ℓ(R×X,E)δ with the inductive limit topology on the right-

hand side.

Remark 4.1.6. Let Y be a Hausdorff-topological vector space. Moreover, let F andG be Frechet spaces which are continuously embedded in Y . Then the non-directsum of the spaces F and G is defined as

F +G := y = f + g ∈ Y ; f ∈ F, g ∈ G,endowed with the following topology: For every continuous seminorm ‖ · ‖F on Fand every continuous seminorm ‖ ·‖G on G define the seminorm ‖ ·‖F+G on F +Gas ‖y‖F+G := inf‖f‖F + ‖g‖G; y = f + g.

Consider the addition + : F⊕G −→ F+G which provides a linear surjection.The kernel is given as ∆ = (f,−f); f ∈ F ∩ G ⊆ Y , and is a closed subspaceof F ⊕ G. The induced mapping on the quotient space (F ⊕ G)/∆ ∼= F + G is atopological isomorphism.

In particular, F + G is a Frechet space, and for Hilbert spaces F and Galso F +G is a Hilbert space (more precisely a hilbertizable space), and we haveF +G ∼= ∆⊥ ⊆ F ⊕G.

Proposition 4.1.7. For s, t, δ ∈ R the following identities hold algebraically andtopologically:

H(s,t);ℓ(R ×X,E)δ =

(〈r〉−δL2(R, Hs+t(X,E))

)∩(〈r〉−δH s

ℓ (R, Ht(X,E)))

for s ≥ 0,(〈r〉−δL2(R, Hs+t(X,E))

)+

(〈r〉−δH s

ℓ (R, Ht(X,E)))

for s ≤ 0.

(4.1.3)

Proof. Without loss of generality assume δ = 0. First we consider the case s ≥ 0.Let u ∈ H(s,t);ℓ(R×X,E) and Rs(τ) ∈ Ls;ℓ(X ; R;E,E) be a parameter-dependentreduction of orders from Theorem 3.1.12. From Theorem 3.1.5 we conclude that

R−s(τ) ∈ S0(R;Ht(X,E), Hs+t(X,E)) ∩ S− sℓ (R;Ht(X,E), Ht(X,E)).

Thus we have

‖F(u)(τ)‖Hs+t(X,E) = ‖R−s(τ)(Rs(τ)F(u)(τ)

)‖Hs+t(X,E)

≤ C‖Rs(τ)F(u)(τ)‖Ht(X,E),

‖F(u)(τ)‖Ht(X,E) = ‖R−s(τ)(Rs(τ)F(u)(τ)

)‖Ht(X,E)

≤ C〈τ〉− sℓ ·‖Rs(τ)F(u)(τ)‖Ht(X,E),


and consequently

H(s,t);ℓ(R ×X,E) → L2(R, Hs+t(X,E))∩H sℓ (R, Ht(X,E)).

Let us show that the embedding is onto. First recall the following elementaryinequality for α, β ∈ C and p > 0:

|α− β|p ≤ max1, 2p−1(|α|p + |β|p

). (1)

Let u ∈ L2(R, Hs(X,E))∩H sℓ (R, L2(X,E)). Passing to local coordinates on X

and E we conclude that(∫

R

∫

Rn

(〈ξ〉2s + 〈τ〉2 sℓ

)‖(Fu)(ξ, τ)‖2

CN−

dξ dτ) 1

2

<∞.

From (1) we see that

〈ξ, τ〉2sℓ ≤ C(〈ξ〉2s + 〈τ〉2 sℓ

)

for (ξ, τ) ∈ Rn × R with a suitable constant C > 0, i. e., locally u belongs to thespace H(s,0);ℓ(Rn+1,CN−). This finishes the proof in the case s ≥ 0 and t = 0.Now let u ∈ L2(R, Hs+t(X,E))∩H s

ℓ (R, Ht(X,E)), and let Λt ∈ Lt(X ;E,E) bea reduction of orders. Then op(Λt)u ∈ L2(R, Hs(X,E))∩H s

ℓ (R, L2(X,E)), i. e.,op(Λt)u ∈ H(s,0);ℓ(R×X,E). Following Seeley’s construction we can arrange that

the reductions of orders Rs(τ) and Λt are commuting, e. g., choose Λt =(C−∆

) t2

and Rs(τ) =(C +

(−∆

)ℓ+ τ2

) s2ℓ with a suitable connection Laplacean ∆ and a

sufficiently large constant C > 0. Then we obtain that u ∈ H(s,t);ℓ(R×X,E) fromDefinition 4.1.2.The case s ≤ 0 follows by duality:Due to Theorem 4.1.5 the space H(s,t);ℓ(R × X,E) equals the dual spaceof H(−s,−t);ℓ(R × X,E) with respect to the sesquilinear pairing induced bythe L2(R, L2(X,E))-inner product. Moreover, we have L2(R, Hs+t(X,E)) ∼=L2(R, H−s−t(X,E))′ and H

sℓ (R, Ht(X,E)) ∼= H− s

ℓ (R, H−t(X,E))′, while thespace S(R×X,E) is dense both in L2(R, H−s−t(X,E)) and H− s

ℓ (R, H−t(X,E)).Thus we obtain the assertion from the already proven result for the spaceH(−s,−t);ℓ(R ×X,E).

Definition 4.1.8. Let ∅ 6= U ⊆ R be an open set.

a) Let H(s,t);ℓ0 (U ×X,E)δ be the subspace of all u ∈ H(s,t);ℓ(R×X,E)δ such that

suppu ⊆ U .

b) Let H(s,t);ℓloc (U × X,E) denote the space of all u ∈ ⋃

s′∈R

D′(U,Hs′(X,E)) such

that for all ϕ ∈ C∞0 (U) the distribution ϕu belongs to H(s,t);ℓ(R × X,E),

endowed with the projective topology with respect to the mappings u 7→ ϕu ∈H(s,t);ℓ(R ×X,E) for all ϕ ∈ C∞

0 (U).

c) Let H(s,t);ℓcomp (U ×X,E) denote the space of all u ∈ H(s,t);ℓ(R ×X,E) such that

suppu ⋐ U is compact. We equip this space with the inductive topology with


respect to the mappings

H(s,t);ℓ(K ×X,E) → H(s,t);ℓcomp (U ×X,E)

for every compact set K ⋐ U , where H(s,t);ℓ(K ×X,E) is the closed subspace

of all u in H(s,t);ℓ(R × X,E) with suppu ⊆ K. Hence H(s,t);ℓcomp (U × X,E) is a

strict (countable) inductive limit.

Theorem 4.1.9. Let ∅ 6= U ⊆ R be an open set.

a) The closure of C∞0 (U,C∞(X,E)) in H(s,t);ℓ(R × X,E)δ is contained in

H(s,t);ℓ0 (U ×X,E)δ.If U is an interval then the closure coincides with the space.

b) H(s,t);ℓloc (U×X,E) is a Frechet space. If V ⊆ R is another open set and χ : U −→

V is a diffeomorphism, then the distributional pull-back χ∗ induces topologicalisomorphisms

χ∗ :

H

(s,t);ℓcomp (V ×X,E) −→ H

(s,t);ℓcomp (U ×X,E),

H(s,t);ℓloc (V ×X,E) −→ H

(s,t);ℓloc (U ×X,E).

4.2. Anisotropic Mellin Sobolev spaces

Remark 4.2.1. Material on isotropic Mellin Sobolev spaces can be found, e. g., in[13], [59], [60], [61].

Notation 4.2.2. For any set Y we denote Y ∧ := R+ × Y .

Definition 4.2.3. For s, t, γ ∈ R the Mellin Sobolev space H(s,t),γ;ℓ(X∧, E) is

defined as the space of all u ∈ ⋃s′∈R

T ′γ−n

2(R+, H

s′(X,E)) such that Mγ−n2u ∈

⋃s′∈R

S′(Γn+12 −γ , H

s′(X,E)) is a regular distribution and

‖u‖H(s,t),γ;ℓ(X∧,E) :=( 1

2πi

∫

Γn+12

−γ

‖Rs(z)(Mγ−n2u)(z)‖2

Ht(X,E) dz) 1

2

<∞.

(4.2.1)Here Rs(z) ∈ Ls;ℓ(X ; Γn+1

2 −γ ;E,E) is a parameter-dependent reduction of orders

from Theorem 3.1.12.

Remark 4.2.4. The space H(s,t),γ;ℓ(X∧, E) is well-defined. From relation (1.1.4) wesee that the transformation Sγ−n

2from (1.1.1) induces a topological isomorphism

Sγ−n2

: H(s,t),γ;ℓ(X∧, E) −→ H(s,t);ℓ(R ×X,E). (4.2.2)

Using (4.2.2) we consequently obtain many properties of the Mellin Sobolev spacesH(s,t),γ;ℓ(X∧, E) from Theorem 4.1.5.

Proposition 4.2.5. a) The relation rδH(s,t),γ;ℓ(X∧, E) = H(s,t),γ+δ;ℓ(X∧, E) isvalid, and we have

H(0,0),0;ℓ(X∧, E) = r−n2 L2(X∧, E) = r−

n2 L2(R+, L

2(X,E)).


More precisely, the following identity holds algebraically and topologically:

H(s,t),γ;ℓ(X∧, E) =

L2,γ−n2 (R+, H

s+t(X,E))∩H sℓ,γ−n

2 (R+, Ht(X,E))

for s ≥ 0,

L2,γ−n2 (R+, H

s+t(X,E)) + H sℓ,γ−n

2 (R+, Ht(X,E))

for s ≤ 0.

(4.2.3)

b) The embeddings H(s,t);ℓcomp (X∧, E) → H(s,t),γ;ℓ(X∧, E) → H

(s,t);ℓloc (X∧, E) are

well-defined and continuous.

Proof. Assertion a) follows from (4.2.2), Proposition 4.1.7 and Definition 2.5.10.b) is a consequence of (4.2.2) and Theorem 4.1.9.

Theorem 4.2.6. a) H(s,t),γ;ℓ(X∧, E) is a Hilbert space with respect to the scalarproduct

〈u, v〉 =1

2πi

∫

Γn+12

−γ

〈Rs(z)(Mγ−n

2u)(z), Rs(z)

(Mγ−n

2v)(z)〉Ht(X,E) dz.

b) The embedding Tγ−n2(X∧, E) = Tγ−n

2(R+, C

∞(X,E)) → H(s,t),γ;ℓ(X∧, E) iscontinuous and dense.

c) The operator of multiplication with a function ϕ ∈ C∞B (R+) induces a contin-

uous operator in L(H(s,t),γ;ℓ(X∧, E)), and the mapping C∞B (R+) ∋ ϕ 7→Mϕ ∈

L(H(s,t),γ;ℓ(X∧, E)) is continuous.

d) For s′−s ≤ min0, t−t′ the embedding H(s,t),γ;ℓ(X∧, E) → H(s′,t′),γ;ℓ(X∧, E)is well-defined and continuous.

e) For k ∈ N0 let Ck;ℓB (R+ × X,E) denote the Banach space of all sectionsu : R+ × X → E such that for jℓ + |α| ≤ k there exists (−r∂r)j∂αx u as abounded continuous function with respect to any choice of local coordinates andtrivializations of the vector bundle, endowed with the topology of uniform con-vergence of all derivatives (−r∂r)j∂αx u up to (anisotropic) order k.

Sobolev embedding theorem: Let k ∈ N0. Then for s > k+ n+ℓ2 the embedding

H(s,0),γ;ℓ(X∧, E) → r−( n+12 −γ)Ck;ℓB (R+ ×X,E) is well-defined and continuous.

f) For every γ0 ∈ R the rγ0−n2 L2(R+ × X,E)-inner product extends to a non-

degenerate sesquilinear pairing

〈·, ·〉γ0 : H(s,t),γ+γ0;ℓ(X∧, E) ×H(−s,−t),−γ+γ0;ℓ(X∧, E) −→ C

which induces an (antilinear) identification of the dual

H(s,t),γ+γ0;ℓ(X∧, E)′ ∼= H(−s,−t),−γ+γ0;ℓ(X∧, E).

Definition 4.2.7. A function ω ∈ C∞0 (R+) such that ω ≡ 1 near r = 0 is called

a cut-off function (near r = 0).

Notation 4.2.8. Let Y be a locally convex space and A ∈ L(Y ). Then we denotethe closure of A(Y ) in Y by [A]Y .


This notation will be employed frequently in case of function spaces Y andmultiplication operators A.

Theorem 4.2.9. Let ω ∈ C∞0 (R+) be a cut-off function near r = 0. Then the

embeddings

[ω]H(s,t),γ;ℓ(X∧, E) → [ω]H(s′,t′),γ′;ℓ(X∧, E),

[1 − ω]H(s,t),γ′;ℓ(X∧, E) → [1 − ω]H(s′,t′),γ;ℓ(X∧, E),

are well-defined and continuous for s′ − s ≤ min0, t− t′ and γ ≥ γ′. Moreover,they are compact if s′ − s < min0, t− t′ and γ > γ′; Hilbert-Schmidt if s′ − s+n+ℓ2 < min0, t− t′ and γ > γ′.

Corollary 4.2.10. Let γ, γ′ ∈ R with γ < γ′. Then for every s, t ∈ R we have

H(s,t),γ;ℓ(X∧, E) ∩H(s,t),γ′;ℓ(X∧, E) =⋂

γ≤δ≤γ′

H(s,t),δ;ℓ(X∧, E).

The intersection is taken in H(s,t);ℓloc (X∧, E).

Definition 4.2.11. Let ∅ 6= U ⊆ R+ be an open set. Define H(s,t),γ;ℓ0 (U ×X,E)

to be the subspace of all u ∈ H(s,t),γ;ℓ(X∧, E) such that suppu ⊆ U .Note that the closure of U is taken with respect to the topology of R+.

Proposition 4.2.12. The closure of C∞0 (U,C∞(X,E)) in H(s,t),γ;ℓ(X∧, E) is

contained in H(s,t),γ;ℓ0 (U × X,E) for every open set ∅ 6= U ⊆ R+. If U is an

interval then the closure coincides with the space.

Mellin Sobolev spaces with asymptotics.

Definition 4.2.13. a) Let −∞ ≤ θ < 0 and Θ := (θ, 0]. For γ ∈ R the tuple(γ,Θ) is called a weight datum.

The strip Γ(n+12 −γ+θ,n+1

2 −γ) ⊆ C is called the weight strip associated with

the weight datum (γ,Θ).b) An asymptotic type associated with the weight datum (γ,Θ) is a finite or count-

ably infinite set

P = (pj,mj , Lj); j ∈ Z (4.2.4)

where the mj ∈ N0 are integers, the Lj are finite-dimensional subspaces ofC∞(X,E) and the pj ∈ C are complex numbers such that with the “projection”πCP := pj ; j ∈ Z of P to C the following properties are fulfilled:

• πCP ⊆ Γ(n+12 −γ+θ,n+1

2 −γ).

• πCP ∩ΓI is finite for every subinterval I ⊆ (n+12 − γ+ θ, n+1

2 − γ) of finitelength.

The collection of all asymptotic types associated with the weight datum (γ,Θ)is denoted by As

((γ,Θ), C∞(X,E)

).


c) Let (γ,Θ) be a weight datum such that θ > −∞. For an asymptotic type Passociated with (γ,Θ) and an arbitrary but fixed cut-off function ω ∈ C∞

0 (R+)near r = 0 we define

EP (X∧, E) := ω(r)∑

(p,m,L)∈P

m∑

k=0

cp,kr−p logk(r); cp,k ∈ L. (4.2.5)

This is a finite-dimensional subspace of C∞(X∧, E), and we endow this spacewith the norm topology.

Definition 4.2.14. Let (γ,Θ) be a weight datum and P ∈ As((γ,Θ), C∞(X,E)

).

a) Define

H(s,t),γ;ℓΘ (X∧, E) :=

⋂

δ∈Θ

H(s,t),γ−δ;ℓ(X∧, E), (4.2.6)

Tγ−n2,Θ(X∧, E) :=

⋂

δ∈Θ

Tγ−n2−δ(X

∧, E), (4.2.7)

endowed with the projective topology with respect to the mappings

H(s,t),γ;ℓΘ (X∧, E) ∋ u 7→ u ∈ H(s,t),γ−δ;ℓ(X∧, E),

Tγ−n2 ,Θ

(X∧, E) ∋ u 7→ u ∈ Tγ−n2 −δ(X

∧, E),

for δ ∈ Θ.Actually, these spaces are Frechet spaces. In (4.2.6) and (4.2.7) we only need

to take the intersection over the elements of a sequence δν; ν ∈ N0 ⊆ Θ suchthat δ0 = 0 and lim

ν→∞δν = θ to obtain them algebraically and topologically, see

also Corollary 4.2.10.b) Let θ > −∞. Define

H(s,t),γ;ℓP (X∧, E) := H(s,t),γ;ℓ

Θ (X∧, E) + EP (X∧, E), (4.2.8)

Tγ−n2,P (X∧, E) := Tγ−n

2,Θ(X∧, E) + EP (X∧, E). (4.2.9)

These spaces are well-defined, i. e., independent of the choice of the particularcut-off function involved in (4.2.5). We equip these spaces with the topology ofthe direct sum which turns them into Frechet spaces.

c) In case of θ = −∞ we define

H(s,t),γ;ℓP (X∧, E) :=

⋂

ν∈N

H(s,t),γ;ℓPν

(X∧, E), (4.2.10)

Tγ−n2 ,P

(X∧, E) :=⋂

ν∈N

Tγ−n2 ,Pν

(X∧, E), (4.2.11)

where the asymptotic type Pν associated with the weight datum (γ, (−ν, 0])contains those elements (p,m,L) ∈ P with p ∈ Γ(n+1

2 −γ−ν,n+12 −γ).

These spaces are Frechet spaces with the projective limit topology inducedby the right-hand sides of (4.2.10), (4.2.11).


Theorem 4.2.15. Let γ, γ′ ∈ R with γ < γ′ and s, t ∈ R. Then for every γ ≤ δ ≤γ′ the weighted Mellin transform

Mδ−n2

: T ′δ−n

2(R+, H

s+t(X,E)) −→ S′(Γn+12 −δ, H

s+t(X,E))

restricts to a topological isomorphism from the intersection H(s,t),γ;ℓ(X∧, E) ∩H(s,t),γ′;ℓ(X∧, E) onto the following space of analytic functions in the stripΓ(n+1

2 −γ′,n+12 −γ):

Let Rs(τ) ∈ Ls;ℓ(X ; R;E,E) be a parameter-dependent reduction of orders

from Theorem 3.1.12. Then a ∈ Mδ−n2

(H(s,t),γ;ℓ(X∧, E) ∩ H(s,t),γ′;ℓ(X∧, E)

)if

and only if

• a ∈ A(Γ(n+12 −γ′,n+1

2 −γ), Hs+t(X,E)),

• ‖a‖ := supγ<δ′<γ′

(1

2πi

∫Γn+1

2−δ′

‖Rs(Im(z))a(z)‖2Ht(X,E) dz

) 12

<∞.

Theorem 4.2.16. Let (γ,Θ) be a weight datum and P ∈ As((γ,Θ), C∞(X,E)

).

a) For every γ ≤ δ < γ − θ the weighted Mellin transform Mδ−n2

restricts to a

topological isomorphism from H(s,t),γ;ℓΘ (X∧, E) onto the space of all

• a ∈ A(Γ( n+12 −γ+θ,n+1

2 −γ), Hs+t(X,E)),

• a|Γ(n+1

2−γ+δ′, n+1

2−γ)

∈ Mγ−n2

(H(s,t),γ;ℓ(X∧, E) ∩ H(s,t),γ−δ′;ℓ(X∧, E)

)for

δ′ ∈ (θ, 0),endowed with the topology of the projective limit.

b) The weighted Mellin transform Mγ−n2

restricts to a topological isomorphismfrom Tγ−n

2,P (X∧, E) onto the following space of meromorphic functions:

a ∈ Mγ−n2

(Tγ−n

2 ,P(X∧, E)

)if and only if:

• a ∈ A(Γ( n+12 −γ+θ,n+1

2 −γ) \ πCP,C∞(X,E)).

• For every (p,m,L) ∈ P we may write in a neighbourhood U(p) \ p

a(z) =

m∑

k=0

νk(z − p)−(k+1) + a0(z)

with νk ∈ L, k = 0, . . . ,m, and a0 holomorphic in p taking values inC∞(X,E).

• For every πCP -excision function χ ∈ C∞(C) the restriction (χ·a)|Γδ be-longs to S(Γδ , C

∞(X,E)), uniformly for δ in subintervals I ⊆ (n+12 − γ+

θ, n+12 − γ) of the form I = [δ′, n+1

2 − γ).This space carries a canonical Frechet topology, namely the convergence of(χ·a)|Γδ ∈ S(Γδ, C

∞(X,E)) for every πCP -excision function χ ∈ C∞(C), uni-formly for δ in subintervals I as above.


c) The following identities hold algebraically and topologically with the topology ofthe non-direct sum of Frechet spaces on the right-hand sides:

H(s,t),γ;ℓP (X∧, E) = H(s,t),γ;ℓ

Θ (X∧, E) + Tγ−n2 ,P

(X∧, E),

Mγ−n2

(H(s,t),γ;ℓP (X∧, E)

)= Mγ−n

2

(H(s,t),γ;ℓ

Θ (X∧, E))

+ Mγ−n2

(Tγ−n

2,P (X∧, E)

),

the latter within meromorphic Hs+t(X,E)-valued functions.

Proof. a) follows immediately from Theorem 4.2.15. Let us prove b). Note firstthat for γ′ ∈ R we have u ∈ Tγ′−n

2(X∧, E) if and only if the function 〈log(r)〉u(r)

belongs to H(s,0),γ′;ℓ(X∧, E) for every s, ∈ R. To see this observe that the trans-formation Sγ′−n

2provides a topological isomorphism

Sγ′−n2

: 〈log(r)〉−H(s,0),γ′;ℓ(X∧, E) −→ H(s,0);ℓ(R ×X,E),

and we have⋂

s,∈R

H(s,0);ℓ(R ×X,E) = S(R ×X,E) by Theorem 4.1.5. Employ-

ing relation (1.1.3) and again Theorem 4.1.5 we obtain assertion b) from a) incase of the empty asymptotic type, i. e., P = Θ. Let us consider general asymp-totic types P . Note that it suffices to prove the assertion for the finite weightinterval, i. e., θ > −∞. Let ω ∈ C∞

0 (R+) be a cut-off function near r = 0, and

v(r) := ω(r)∑

(p,m,L)∈P

m∑k=0

cp,kr−p logk(r) ∈ EP (X∧, E) with cp,k ∈ L. Then, by the

properties of the Mellin transform, we have

(Mγ−n2v)(z) =

∑

(p,m,L)∈P

m∑

k=0

cp,k

( d

dz

)k( 1

z − pM(−r∂rω)(z − p)

)(1)

for z ∈ C, and consequently the asserted characterizations hold for the functions inEP (X∧, E). Summing up, we have proved that the Mellin transform of a functionin Tγ−n

2,P (X∧, E) is meromorphic in the weight strip with the desired properties.

Conversely, let a be a meromorphic function in the weight strip with the propertieslisted in b). Then we see from (1) that there is a function v ∈ EP (X∧, E) suchthat a(z) − (Mγ−n

2v)(z) is holomorphic and satisfies the conditions in b) for the

empty asymptotic type, i. e., there is a function u ∈ Tγ−n2,Θ(X∧, E) such that

(Mγ−n2u)(z) = a(z) − (Mγ−n

2v)(z). Thus we have a = Mγ−n

2(u + v) which

finishes the proof of b).Clearly, the assertions in c) are equivalent by Mellin transform. For the finite

weight interval we have nothing to prove, so let us consider the infinite interval

Θ = (−∞, 0]. For H(s,t),γ;ℓP (X∧, E) ⊇ H(s,t),γ;ℓ

Θ (X∧, E)+Tγ−n2 ,P

(X∧, E) is obvious

we only have to check the opposite inclusion. Let a ∈ Mγ−n2

(H(s,t),γ;ℓP (X∧, E)

)

be arbitrary, and let ω ∈ C∞0 (R+) be a cut-off function near r = 0. Let

qj; j ∈ N0 ⊆ πCP be the pole pattern of the meromorphic function a. For


every pole qj ∈ C with (qj ,mj , Lj) ∈ P choose functions cqj ,k ∈ Lj such that

a(z)−Mγ−n2

(ω(r)

mj∑k=0

cqj ,kr−qj logk(r)

)(z) is holomorphic in qj . For c > 0 define

ψc,qj (z) := Mγ−n2

(ω(cr)

mj∑

k=0

cqj ,kr−qj logk(r)

)(z).

Then also a(z) − ψc,qj (z) is holomorphic in qj by (1). Note that for c > 0 wehave M(−r∂rω(cr))(z) = c−zM(−r∂rω)(z), and consequently M(−r∂rω(cr))converges to 0 in S(Γβ) as c → ∞, locally uniformly for β in R+. A Borel ar-gument now shows that there is a sequence (cj) ⊆ R+ with lim

j→∞cj = ∞ such that

the series

b(z) :=∞∑

j=0

ψcj ,qj (z)

converges and defines an element b ∈ Mγ−n2

(Tγ−n

2 ,P(X∧, E)

)in view of b). More-

over, a − b belongs to Mγ−n2

(H(s,t),γ;ℓ

(−∞,0] (X∧, E))

by a) which finishes the proof of

the theorem.

4.3. Cone Sobolev spaces

Remark 4.3.1. In this section we introduce anisotropic Sobolev spaces onX∧ whichcoincide near r = 0 with the Mellin Sobolev spaces from Section 4.2, and nearr = ∞ with the Sobolev spaces from Section 4.1. The construction is analogous tothat of the cone Sobolev spaces considered in the analysis on spaces with conicalsingularities, cf. [13], [59], [60], [61], which motivates the name and the notationsinvolved. Nevertheless, even in the isotropic case, i. e., ℓ = 1 and t = 0, the spacesdiffer from each other near r = ∞: While the “classical” cone Sobolev spacesreflect the conical structure near infinity in polar coordinates, the spaces from thissection impose the structure of a cylindrical end.

Our main interest in this part is the analysis of parabolic pseudodifferentialoperators and the behaviour of solutions on the closed compact manifold X . Thespace–time configuration for these problems is [t0,∞)×X for some t0 ∈ R, wheret = ∞ is treated as a cylindrical end with an exponential weight. In our approach,this configuration is transformed via (1.1.1) to (0, r0] × X with r0 ∈ R+, andthe corresponding function spaces are the (weighted) Mellin Sobolev spaces fromSection 4.2. Consequently, for the applications we have in mind, the particularchoice of the function space onX∧ near r = ∞ is irrelevant as far as it is compatiblewith the Mellin Sobolev space away from infinity.

The analysis of the operators within the cone Sobolev spaces from this sectionturns out to be quite natural in view of the examples involved in the applications,which motivates the definitions and constructions given below.

Definition 4.3.2. Let ω ∈ C∞0 (R+) be a cut-off function near r = 0.


a) For γ ∈ R define

Sγ(X∧, E) := [ω]Tγ−n2(X∧, E) + [1 − ω]S(R ×X,E). (4.3.1)

b) For s, t, γ, δ ∈ R define the cone Sobolev space K(s,t),γ;ℓ(X∧, E)δ as

K(s,t),γ;ℓ(X∧, E)δ := [ω]H(s,t),γ;ℓ(X∧, E) + [1 − ω]H(s,t);ℓ(R ×X,E)δ. (4.3.2)

The non-direct sums are carried out in H(s,t);ℓloc (X∧, E), and the resulting spaces

do not depend on the particular choice of the cut-off function.

Notation 4.3.3. For γ, δ ∈ R let kγ,δ ∈ C∞(R+) be an everywhere positive functionwith

kγ,δ ≡rγ−

n2 near r = 0

r−δ near r = ∞.

Theorem 4.3.4. Let ω ∈ C∞0 (R+) be a cut-off function.

a) Sγ(X∧, E) is a nuclear Frechet space.b) K(s,t),γ;ℓ(X∧, E)δ is a Hilbert space (more precisely a hilbertizable space).

c) We have kγ′+n2,δ′K(s,t),γ;ℓ(X∧, E)δ = K(s,t),γ+γ′;ℓ(X∧, E)δ+δ′ algebraically and

topologically. Moreover, we have K(0,0),γ;ℓ(X∧, E)δ = kγ,δL2(X∧, E) and in

particular K(0,0),0;ℓ(X∧, E)n2

= r−n2 L2(X∧, E) (see also i) below).

d) The embedding Sγ(X∧, E) → K(s,t),γ;ℓ(X∧, E)δ is continuous and dense.e) The operator of multiplication with a function ϕ ∈ [ω]C∞

B (R+)+ [1−ω]C∞b (R)

induces a continuous operator in L(K(s,t),γ;ℓ(X∧, E)δ), and the mapping

[ω]C∞B (R+) + [1 − ω]C∞

b (R) ∋ ϕ 7→Mϕ ∈ L(K(s,t),γ;ℓ(X∧, E)δ)

is continuous.f) For s′−s ≤ min0, t−t′, γ ≥ γ′ and δ ≥ δ′ the embedding K(s,t),γ;ℓ(X∧, E)δ →

K(s′,t′),γ′;ℓ(X∧, E)δ′ is well-defined and continuous. Moreover, it is compact ifs′ − s < min0, t − t′, γ > γ′ and δ > δ′; Hilbert-Schmidt if s′ − s + n+ℓ

2 <

min0, t− t′, γ > γ′ and δ − δ′ > 12 .

g) Sobolev embedding theorem: Let k ∈ N0. Then for s > k + n+ℓ2 the embedding

K(s,0),γ;ℓ(X∧, E)δ → [ω]r−( n+12 −γ)Ck;ℓB (R+ ×X,E) + [1 − ω]〈r〉−δCk;ℓb (R ×X,E)

is well-defined and continuous.h) For every δ0, γ0 ∈ R the kγ0,δ0L

2(X∧, E)-inner product extends to a non-degenerate sesquilinear pairing

〈·, ·〉γ0,δ0 : K(s,t),γ+γ0;ℓ(X∧, E)δ0+δ ×K(−s,−t),−γ+γ0;ℓ(X∧, E)δ0−δ −→ C

which induces an identification of the dual

K(s,t),γ+γ0;ℓ(X∧, E)′δ0+δ∼= K(−s,−t),−γ+γ0;ℓ(X∧, E)δ0−δ.

The r−n2 L2(X∧, E)-inner product 〈·, ·〉 serves as the reference inner product in

the scale(K(s,t),γ;ℓ(X∧, E)δ

)s,t,γ,δ∈R

.


i) For s, t, γ, δ ∈ R the following identities hold algebraically and topologically:

K(s,t),γ;ℓ(X∧, E)δ =

kγ,δL2(R, Hs+t(X,E))∩K s

ℓ,γ−n

2 (R+, Ht(X,E))δ

for s ≥ 0,

kγ,δL2(R, Hs+t(X,E)) + K s

ℓ,γ−n

2 (R+, Ht(X,E))δ

for s ≤ 0,

(4.3.3)where

K sℓ,γ−n

2 (R+, Ht(X,E))δ :=[ω]H s

ℓ,γ−n

2 (R+, Ht(X,E))+

[1 − ω]〈r〉−δH sℓ (R, Ht(X,E)).

Proof. a) and b) are consequences of Remark 4.1.6 and the permanence propertiesof nuclear spaces. c)–e), g) and h) follow from Theorem 4.1.5, Theorem 4.2.6 andProposition 4.2.5. f) is a consequence of Theorem 4.1.5 and Theorem 4.2.9. i)follows from Proposition 4.1.7 and Proposition 4.2.5.

Remark 4.3.5. Let

A : K(s,t),γ;ℓ(X∧, E)δ −→ K(s−µ,t),γ;ℓ(X∧, F )δ−

be continuous for all s, t, δ ∈ R. Then the formal adjoint operator A∗ with respectto the r−

n2 L2-inner product 〈·, ·〉 is defined by means of the identity 〈Au, v〉 =

〈u,A∗v〉. By Theorem 4.3.4 the operatorA∗ is well-defined as a continuous operator

A∗ : K(s,t),−γ;ℓ(X∧, F )δ −→ K(s−µ,t),−γ;ℓ(X∧, E)δ−

for all s, t, δ ∈ R. In the remaining part we will take formal adjoints of operatorson X∧ in this sense.

Definition 4.3.6. Let ∅ 6= U ⊆ R+ be an open set. Define K(s,t),γ;ℓ0 (U ×X,E)δ

to be the subspace of all u ∈ K(s,t),γ;ℓ(X∧, E)δ such that suppu ⊆ U .The closure of U is taken with respect to the topology of R+.

Proposition 4.3.7. The closure of C∞0 (U,C∞(X,E)) in K(s,t),γ;ℓ(X∧, E)δ is con-

tained in K(s,t),γ;ℓ0 (U ×X,E)δ for every open set ∅ 6= U ⊆ R+. If U is an interval

then the closure coincides with the space. Moreover, the following identities arevalid:

K(s,t),γ;ℓ0 ((0, r0] ×X,E)δ = H(s,t),γ;ℓ

0 ((0, r0] ×X,E),

K(s,t),γ;ℓ0 ([r0,∞) ×X,E)δ = H

(s,t);ℓ0 ([r0,∞) ×X,E)δ

for every r0 ∈ R+.

Proof. These assertions follow from Theorem 4.1.9 and Proposition 4.2.12.


Definition 4.3.8. Let (γ,Θ) be a weight datum and P ∈ As((γ,Θ), C∞(X,E)

).

For an arbitrary but fixed cut-off function ω ∈ C∞0 (R+) near r = 0 define

SγP (X∧, E) := [ω]Tγ−n2,P (X∧, E) + [1 − ω]S(R ×X,E),

K(s,t),γ;ℓP (X∧, E)δ := [ω]H(s,t),γ;ℓ

P (X∧, E) + [1 − ω]H(s,t);ℓ(R ×X,E)δ,

for s, t, δ ∈ R. These spaces are independent of the particular choice of the cut-offfunction ω ∈ C∞

0 (R+), and we endow them with the (Frechet) topology of thenon-direct sum.

Proposition 4.3.9. a) The embeddings

SγP (X∧, E) → Sγ(X∧, E) → K(s,t),γ;ℓ(X∧, E)δ,

SγP (X∧, E) → K(s,t),γ;ℓP (X∧, E)δ → K(s,t),γ;ℓ(X∧, E)δ

are well-defined and continuous for every s, t, δ ∈ R.b) If θ > −∞ we have direct decompositions

SγP (X∧, E) = SγΘ(X∧, E) + EP (X∧, E),

K(s,t),γ;ℓP (X∧, E)δ = K(s,t),γ;ℓ

Θ (X∧, E)δ + EP (X∧, E).

c) We have

SγP (X∧, E) =⋂

s,δ∈R

K(s,0),γ;ℓP (X∧, E)δ,

and SγP (X∧, E) is a nuclear Frechet space.

Proof. a) and b) are obvious. We have to prove the representation of SγP (X∧, E) in

c) as the intersection over the K(s,0),γ;ℓP (X∧, E)δ–spaces in case of the finite weight

interval only. From b) we conclude that it suffices to consider the empty asymp-totic type. But then the desired identity holds in view of the Sobolev embeddingtheorem in Theorem 4.3.4. Let us prove the nuclearity of the space SγP (X∧, E).By the permanence properties of nuclear spaces we just have to consider the finiteweight interval and the empty asymptotic type. By the closed graph theorem therepresentation in c) holds topologically with the projective limit topology on theright-hand side. Employing the embedding properties from Theorem 4.3.4 f) weget the asserted nuclearity.

Chapter 5. Calculi built upon parameter-dependent operators

5.1. Anisotropic meromorphic Mellin symbols

Remark 5.1.1. In this chapter we shall again employ the notations from Notation3.1.1 with the corresponding data fixed on X and the vector bundles E and F .


Lemma 5.1.2. Let ∅ 6= I ⊆ R be an open interval and µ ∈ R. Let

ℓ∞locAµ;ℓ(cl) := a ∈ A(ΓI , L

µ(X ;E,F )); a|Γβ ∈ Lµ;ℓ(cl)(X ; Γβ;E,F )

locally uniformly for β ∈ IC∞Aµ;ℓ

(cl) := a ∈ A(ΓI , Lµ(X ;E,F )); a ∈ C∞(Iβ , L

µ;ℓ(cl)(X ; Γβ;E,F ))

endowed with their natural Frechet topologies.

Then the embedding ι : C∞Aµ;ℓ(cl) → ℓ∞locAµ;ℓ

(cl) is onto and provides an isomor-

phism between these spaces. The complex derivative acts linear and continuous in

the spaces ∂z : ℓ∞locAµ;ℓ(cl) → ℓ∞locAµ−ℓ;ℓ

(cl) .

Moreover, given a ∈ ℓ∞locAµ;ℓ(cl), we have the following asymptotic expansion for

a|Γβ0 in terms of a|Γβ for every β0, β ∈ I which depends smoothly on (β0, β) ∈ I×I:

a|Γβ0 ∼∞∑

k=0

(β0 − β)k

k!

(∂kz a

)|Γβ .

Proof. Passing via (3.1.4) to local symbols and global remainders on X revealsthat the assertions follow from Proposition 2.6.3.

Notation 5.1.3. For p ∈ C and k ∈ N0 let ψp,k ∈ A(C\p) be an analytic functionwhich is meromorphic in p with a pole of multiplicity k + 1 such that for everyp-excision function χ ∈ C∞(C) the function χ · ψp,k belongs to C∞(Rβ ,S(Γβ)).

In view of the properties of the Mellin transform the function

ψp,k(z) := Mγ(ω(r)r−p logk r)(z) =( d

dz

)k( 1

z − p·M(−r∂rω)(z − p)

),

where ω ∈ C∞0 (R+) is a cut-off function near r = 0 and γ < 1

2 − Re(p), fulfillsthese conditions.

Definition 5.1.4. A Mellin asymptotic type is a finite or countably infinite set

P = (pj,mj , Lj); j ∈ Z (5.1.1)

where the mj ∈ N0 are integers, the Lj are finite-dimensional subspaces ofL−∞(X ;E,F ) consisting of finite-dimensional operators, and the pj ∈ C are com-plex numbers such that with the “projection” πCP := pj; j ∈ Z of P to C wehave that the set πCP ∩ΓI is finite for every compact interval I ⊆ R. For the emptyasymptotic type we use the notation O. The collection of all Mellin asymptotictypes is denoted by As

(L−∞(X ;E,F )

).

Definition 5.1.5. For µ ∈ R and P ∈ As(L−∞(X ;E,F )

)we define the space

Mµ;ℓP (cl)(X ;E,F ) of (anisotropic) meromorphic Mellin symbols of order µ with as-

ymptotic type P to consist of all functions a ∈ A(C \ πCP,Lµ(X ;E,F )) with the

following properties:


• For every (p,m,L) ∈ P we may write in a neighbourhood U(p) \ p

a(z) =

m∑

k=0

νk(z − p)−(k+1) + a0(z)

with νk ∈ L, k = 0, . . . ,m, and a0 holomorphic in p taking values inLµ(X ;E,F ).

• For every compact interval I ⊆ R we have

a(β + iτ) −∑

(pj ,mj,Lj); Re(pj)∈I

mj∑

k=0

σjkψpj ,k ∈ Lµ;ℓ(cl)(X ; Γβ;E,F ) (5.1.2)

uniformly for β ∈ I with suitable σjk ∈ Lj .

Analogously, we define the space M−∞P (X ;E,F ) of meromorphic Mellin symbols

of order −∞ with asymptotic type P .If P = O is the empty asymptotic type the spaces are called holomorphic

Mellin symbols.

Remark 5.1.6. The topology on the space Mµ;ℓP (cl)(X ;E,F ) is determined by the

following ingredients:

• The topology of A(C \ πCP,Lµ(X ;E,F )).

• Convergence of the Laurent coefficients νk in the corresponding coefficientspaces Lj ⊆ L−∞(X ;E,F ).

• Uniform convergence of (5.1.2) for β ∈ I for every compact interval I ⊆ R.

With this topology Mµ;ℓP (cl)(X ;E,F ) is a Frechet space. Note that the topology

does not depend on the particular choice of the functions ψpj ,k from Notation5.1.3 involved in (5.1.2) and the coefficients σjk determined by them in view of theclosed graph theorem.

In order −∞ we have an equivalent characterization of M−∞P (X ;E,F ) as the

space of all analytic functions a ∈ A(C \ πCP,L−∞(X ;E,F )) such that

• a is meromorphic in p for every (p,m,L) ∈ P , and in a neighbourhoodU(p) \ p we have

a(z) =

m∑

k=0

νk(z − p)−(k+1) + a0(z)

with νk ∈ L, k = 0, . . . ,m, and a0 holomorphic in p taking values inL−∞(X ;E,F ).

• We have χ·a ∈ C∞(Rβ , L−∞(X ; Γβ;E,F )) for every πCP -excision func-

tion χ ∈ C∞(C).

If we choose a countable collection of πCP -excision functions χkk ⊆ C∞(C)which shrink to πCP we find that the Frechet space structure on M−∞

P (X ;E,F )is determined by the projective topology with respect to the mappings

M−∞P (X ;E,F ) ∋ a 7−→ χk·a ∈ C

(Rβ, L

−∞(X ; Γβ;E,F )).

Note that Lemma 5.1.2 enters these considerations.


Material on (scalar) isotropic meromorphic Mellin symbols, i. e., ℓ = 1, canbe found, e. g., in [13], [59], [60], [61].

Proposition 5.1.7. Let µ, µ′ ∈ R, µ′ ≤ µ. Moreover, let P ∈ As(L−∞(X ;E,F )

)

and β ∈ R with Γβ ∩ πCP = ∅. The following identities hold algebraically andtopologically:

a) Mµ;ℓP (X ;E,F ) ∩ Lµ′;ℓ(X ; Γβ;E,F ) = Mµ′;ℓ

P (X ;E,F ).

b) Mµ;ℓP (X ;E,F ) ∩ Lµ;ℓ

cl (X ; Γβ;E,F ) = Mµ;ℓP cl(X ;E,F ).

For holomorphic Mellin symbols we have:

i) Mµ;ℓO(cl)(X ;E,F ) = A(C, Lµ(X ;E,F )) ∩C∞(Rβ , L

µ;ℓ(cl)(X ; Γβ;E,F )).

ii) The complex derivative acts continuous in the spaces

∂z : Mµ;ℓO(cl)(X ;E,F ) →Mµ−ℓ;ℓ

O(cl) (X ;E,F ).

iii) For a ∈ Mµ;ℓO(cl)(X ;E,F ) the following asymptotic expansion holds for a|Γβ0

in terms of a|Γβ which depends smoothly on (β0, β) ∈ R × R:

a|Γβ0 ∼∞∑

k=0

(β0 − β)k

k!

(∂kz a

)|Γβ .

In the classical case we consequently obtain for arbitrary β0, β ∈ R the follow-ing relationship for the parameter-dependent homogeneous principal symbol:

σµ;ℓψ (a|Γβ0 ) = σµ;ℓ

ψ (a|Γβ ).iv) For s, ν ∈ R with ν ≥ µ we have a continuous embedding

Mµ;ℓO (X ;E,F ) →

Sµℓ

O (C;Hs(X,E), Hs−ν(X,F )) ν ≥ 0

Sµ−νℓ

O (C;Hs(X,E), Hs−ν(X,F )) ν ≤ 0.

Proof. The assertions i) – iii) are subject to Lemma 5.1.2, while iv) follows fromi) and Theorem 3.1.5.

To prove a) let a ∈ Mµ;ℓP (X ;E,F ) ∩ Lµ′;ℓ(X ; Γβ;E,F ). Consider for N ∈ N

such that N > |β| the open interval I := (−N,N) ⊆ R. According to Defini-tion 5.1.5 of meromorphic Mellin symbols we conclude from Lemma 5.1.2 that theexpression (5.1.2) belongs to the space ℓ∞locAµ;ℓ over the interval I. From the as-ymptotic expansion result in Lemma 5.1.2 and the assumption that a is of order µ′

on the weight line Γβ we even conclude that (5.1.2) belongs to the space ℓ∞locAµ′;ℓ

over I. But sinceN ∈ N withN > |β| was arbitrary we see that a ∈Mµ′;ℓP (X ;E,F )

as asserted. This shows that the identity in a) holds algebraically, but then it holdsalso topologically in view of the closed graph theorem. The same reasoning as inthe proof of a) also yields assertion b).

Theorem 5.1.8. a) Let P,Q ∈ As(L−∞(X ;E,F )

). Then (pointwise) addition as

Lµ(X ;E,F )-valued operator functions on C \(πCP ∪ πCQ

)induces a bilinear


and continuous mapping

+ : Mµ;ℓP (cl)(X ;E,F ) ×Mµ;ℓ

Q(cl)(X ;E,F ) −→Mµ;ℓR(cl)(X ;E,F ),

where R ∈ As(L−∞(X ;E,F )

)consists (in general) of elements (q,m,L) of the

form

(q,maxm1,m2, L1 + L2) if (q,m1, L1) ∈ P and (q,m2, L2) ∈ Q

(q,m,L) if q ∈ πCPπCQ and (q,m,L) ∈ P ∪Q.

b) Let G be another vector bundle over X and a ∈ Mµ;ℓP (cl)(X ;G,F ), b ∈

Mµ′;ℓQ(cl)(X ;E,G). Then the pointwise composition (multiplication) as functions

on C \(πCP ∪ πCQ

)gives rise to an element ab ∈ Mµ+µ′;ℓ

R(cl) (X ;E,F ) with a

resulting asymptotic type R ∈ As(L−∞(X ;E,F )

)which consists (in general)

of elements (q,m,L) of the form

(q,m1 +m2 + 1, L) if (q,m1, L1) ∈ P and (q,m2, L2) ∈ Q

(q,m,L) if q ∈ πCPπCQ and (q,m, L) ∈ P ∪Q.c) For holomorphic Mellin symbols the multiplication as functions on C gives rise

to a continuous bilinear mapping

· : Mµ;ℓO(cl)(X ;G,F ) ×Mµ′;ℓ

O(cl)(X ;E,G) −→Mµ+µ′;ℓO(cl) (X ;E,F ).

Proof. These assertions follow from the Definition 5.1.5 of meromorphic Mellinsymbols and the properties of anisotropic parameter-dependent pseudodifferentialoperators on closed compact manifolds as discussed in the Section 3.1 (for thecomposition note in particular Theorem 3.1.6).

Remark 5.1.9. Let µ ∈ R and µ+ := max0, µ. Then the Mellin kernel cut-offoperator with respect to the weight γ ∈ R is bilinear and continuous in the spaces

Hγ : C∞B (R+)×S µ

ℓ (Γ 12−γ

;Hs(X,E), Hs−µ+(X,F ))

−→ Sµℓ (Γ 1

2−γ;Hs(X,E), Hs−µ+(X,F ))

by Theorem 2.6.13. Analogously to Theorem 3.2.9 and Corollary 3.2.10 we obtainthe following theorem for the Mellin kernel cut-off operator from Theorem 2.6.13.

Theorem 5.1.10. The Mellin kernel cut-off operator with respect to the weightγ ∈ R restricts to continuous bilinear mappings in the spaces

Hγ :

C∞B (R+)×Lµ;ℓ

(cl)(X ; Γ 12−γ

;E,F ) −→ Lµ;ℓ(cl)(X ; Γ 1

2−γ;E,F )

C∞0 (R+)×Lµ;ℓ

(cl)(X ; Γ 12−γ

;E,F ) −→ Mµ;ℓO(cl)(X ;E,F ).

Given ϕ ∈ C∞B (R+) and a ∈ Lµ;ℓ(X ; Γ 1

2−γ;E,F ) we have the following asymptotic

expansion of Hγ(ϕ)a in terms of ϕ and a in the sense of Definition 3.1.8:

Hγ(ϕ)a ∼∞∑

k=0

1

k!(r∂r)

kϕ(r)|r=1 ·Dkτa.


For ψ ∈ C∞0 (R+) such that ψ ≡ 1 near r = 1 the operator I−Hγ(ψ) is continuous

in the spaces

I −Hγ(ψ) : Lµ;ℓ(X ; Γ 12−γ

;E,F ) −→ L−∞(X ; Γ 12−γ

;E,F ).

Corollary 5.1.11. For P ∈ As(L−∞(X ;E,F )

)we have

Mµ;ℓP (cl)(X ;E,F ) = Mµ;ℓ

O(cl)(X ;E,F ) +M−∞P (X ;E,F )

algebraically and topologically with the topology of the non-direct sum of Frechetspaces on the right-hand sides.

Proof. Let ψ ∈ C∞0 (R+) such that ψ ≡ 1 near r = 1. Let γ ∈ R such that

Γ 12−γ

∩ πCP = ∅. In view of Theorem 5.1.10 and Theorem 5.1.8 we may write for

a ∈Mµ;ℓP (cl)(X ;E,F )

a = Hγ(ψ)a+(I −Hγ(ψ)

)a,

where Hγ(ψ)a ∈ Mµ;ℓO(cl)(X ;E,F ), and

(I − Hγ(ψ)

)a belongs to the space

L−∞(X ; Γ 12−γ

;E,F )∩Mµ;ℓP (cl)(X ;E,F ) = M−∞

P (X ;E,F ) due to Proposition 5.1.7

and Theorem 5.1.8. This provides a topological isomorphism as asserted.

Definition 5.1.12. Let P ∈ As(L−∞(X ;E,F )

). A meromorphic Mellin symbol

a ∈ Mµ;ℓP (cl)(X ;E,F ) is called elliptic, if the restriction a|Γβ ∈ Lµ;ℓ

(cl)(X ; Γβ;E,F )

to some weight line Γβ is parameter-dependent elliptic in the sense of Definition3.1.9, where β ∈ R is such that Γβ ∩ πCP = ∅.

According to Corollary 5.1.11 and Proposition 5.1.7 this is well-defined inthe sense that for every β ∈ R such that Γβ ∩ πCP = ∅ the restriction a|Γβ ∈Lµ;ℓ

(cl)(X ; Γβ;E,F ) is parameter-dependent elliptic if and only if it is the case for

some β ∈ R.

Proposition 5.1.13. Let c ∈ M−∞P (X ;E,E). Then there exists an element d ∈

M−∞Q (X ;E,E) such that

(1 + c

)−1= 1 + d as meromorphic operator functions.

Proof. First observe that the function 1+ c ∈ A(C\πCP,L0(X ;E,E)) is a finitely

meromorphic Fredholm family taking values in L0(X ;E,E) → L(L2(X,E)).Let β ∈ R such that Γβ ∩ πCP = ∅. Then

(1 + c

)|Γβ ∈ L0;ℓ(X ; Γβ;E,E) is

parameter-dependent elliptic. Thus, by virtue of Theorem 3.1.11, there exists(1 + c(z)

)−1 ∈ L0(X ;E,E) for |Im(z)| sufficiently large on Γβ . Consequently,we may apply Theorem 1.2.6 on the inversion of finitely meromorphic Fredholmfamilies to 1 + c, i. e., 1 + c is invertible as a finitely meromorphic Fredholmfamily (taking values in L0(X ;E,E)). The Laurent-coefficients of the principal

part of(1 + c

)−1at a pole p ∈ C are finite-dimensional pseudodifferential op-

erators, and thus they necessarily belong to L−∞(X ;E,E). Moreover, we have(1 + c

)−1= 1 − c + c

(1 + c

)−1c, which shows that d := −c + c

(1 + c

)−1c is a

meromorphic function on C taking values in L−∞(X ;E,E).


Let us study the inverse(1 + c

)−1in more detail. Let R > 0 be arbitrary

such that(Γ−R ∪ ΓR

)∩ πCP = ∅. Let χ ∈ C∞(C) such that χ ≡ 0 near πCP ∩

Γ[−R,R] and χ ≡ 1 outside some small neighbourhood U of πCP ∩ Γ[−R,R]. Thenin view of Definition 5.1.5 and Remark 5.1.6 the function χ·c depends smoothlyon β ∈ [−R,R] taking values in L−∞(X ; Γβ;E,E). Now apply Theorem 3.1.11 to1 + χ·c. This shows at first that for |Im(z)| sufficiently large in Γ[−R,R] we have

that 1 +χ(z)c(z) is invertible. Moreover, we have(1 +χ(z)c(z)

)−1=

(1 + c(z)

)−1

outside U . But since the neighbourhood U (i. e., the excision function χ) may be

chosen arbitrarily small we conclude that only finitely many poles of(1+ c

)−1are

located in the strip Γ[−R,R]. Consequently, the pattern of poles together with the

data of the Laurent expansions of(1 + c

)−1determines a Mellin asymptotic type

Q ∈ As(L−∞(X ;E,E)

).

Now let R > 0 be arbitrary such that(Γ−R ∪ ΓR

)∩

(πCP ∪ πCQ

)= ∅. We

have that 1+ c(z) is invertible for z ∈ Γ[−R,R] \(πCP ∪πCQ

). Let V be some small

neighbourhood of(πCP ∪ πCQ

)∩ Γ[−R,R]. Choose χ ∈ C∞(C) such that χ ≡ 0

near(πCP ∪ πCQ

)∩ Γ[−R,R] and χ ≡ 1 outside V . For z ∈ Γ[−R,R] \ V we may

write using Theorem 3.1.11

1 + d(z) =(1 + c(z)

)−1=

(1 + χ(z)c(z)

)−1= 1 + c(z)

where c ∈ C∞([−R,R], L−∞(X ; Γβ;E,E)). This shows that d ∈ M−∞Q (X ;E,E)

which finishes the proof of the proposition.

Theorem 5.1.14. An element a ∈Mµ;ℓP (cl)(X ;E,F ) is elliptic if and only if there

exists b ∈M−µ;ℓQ(cl)(X ;F,E) such that a·b ≡ 1 and b·a ≡ 1, i. e., a is invertible as a

meromorphic operator function with a−1 = b ∈M−µ;ℓQ(cl)(X ;F,E).

Proof. Let a ∈ Mµ;ℓP (cl)(X ;E,F ) be elliptic and γ ∈ R such that Γ 1

2−γ∩ πCP = ∅.

According to Theorem 3.1.10 there exists p ∈ L−µ;ℓ(cl) (X ; Γ 1

2−γ;F,E) such that

(a|Γ 12−γ

)·p− 1 ∈ L−∞(X ; Γ 12−γ

;F, F ) and p·(a|Γ 12−γ

) − 1 ∈ L−∞(X ; Γ 12−γ

;E,E).

Let ψ ∈ C∞0 (R+) with ψ ≡ 1 near r = 1 and define b := Hγ(ψ)p. Using Theorem

5.1.10 we get b ∈ M−µ;ℓO(cl)(X ;F,E) and

(p − b

)|Γ 1

2−γ

∈ L−∞(X ; Γ 12−γ

;F,E). This

shows that b|Γ 12−γ

is a parameter-dependent parametrix of a|Γ 12−γ

. Moreover, from

Theorem 5.1.8 and Proposition 5.1.7 we obtain that a·b = 1 + rR and b·a = 1 + rLwith meromorphic Mellin symbols rL and rR of order −∞. Now apply Proposition5.1.13 to 1 + rL and 1 + rR. Then we conclude from Theorem 5.1.8 that

a−1 = b =(1 + rL

)−1b = b

(1 + rR

)−1 ∈M−µ;ℓQ(cl)(X ;F,E)

as asserted. If conversely a is invertible as a meromorphic Mellin symbol with

inverse b ∈M−µ;ℓQ(cl)(X ;F,E) we see that b|Γβ is a parameter-dependent parametrix


of a|Γβ for every β ∈ R such that Γβ∩(πCP ∪πCQ

)= ∅. Consequently, a is elliptic

in the sense of Definition 5.1.12 due to Theorem 3.1.10.

Theorem 5.1.15. Let I ⊆ R be a compact interval. Then there exists for

every µ ∈ R an elliptic element h ∈ Mµ;ℓO (X ;E,E) such that its inverse

h−1 ∈ M−µ;ℓQ (X ;E,E) (cf. Theorem 5.1.14) has no poles in the strip ΓI , i. e.,

ΓI∩πCQ = ∅.Proof. According to Theorem 3.1.12 there exist for µ ∈ R operators aµ ∈Lµ;ℓ(X ; Γ0×Rλ;E,E) such that aµa−µ = 1. Now let µ ∈ R be given. Letϕ ∈ C∞

0 (R+) such that ϕ ≡ 1 near r = 1. Define for (z, λ) ∈ Γ0×R

a(z, λ) :=(H 1

2(ϕ)aµ

)(z, λ).

From Theorem 5.1.10 and Theorem 2.6.13 we conclude that a(z, λ) gives rise to a

parameter-dependent family in Mµ;ℓO (X ;E,E) depending on the parameter λ ∈ R.

More precisely, we have for β ∈ R

a(·, λ)|Γβ =(H 1

2(ϕ)aµ

)|Γβ (λ) =

(H 1

2(rβϕ)aµ

)|Γ0(λ).

For the family rβϕ(r); β ∈ I ⊆ C∞0 (R+) is bounded we conclude from Theorem

5.1.10 and Theorem 3.1.11 that if we fix λ0 ∈ R with |λ0| sufficiently large we canarrange the invertibility of a(z, λ0) : Hs(X,E) → Hs−µ(X,E) for all z ∈ ΓI . Thus

the symbol h := a(z, λ0) ∈ Mµ;ℓO (X ;E,E) has the desired properties in view of

Theorem 5.1.14.

5.2. Meromorphic Volterra Mellin symbols

Definition 5.2.1. Let Hβ be a right half-plane in C. For µ ∈ R and a Mellinasymptotic type P ∈ As

(L−∞(X ;E,F )

)such that πCP ∩ Hβ = ∅ we define the

space of meromorphic Volterra Mellin symbols of order µ with asymptotic type Pas

Mµ;ℓV,P (cl)(X ; Hβ;E,F ) := Mµ;ℓ

P (X ;E,F )∩Lµ;ℓV (cl)(X ; Hβ;E,F )

with the induced Frechet topology.Analogously, we define the space M−∞

V,P (X ; Hβ;E,F ) of meromorphic Mellinsymbols of order −∞ with asymptotic type P . If P = O is the empty asymptotictype the spaces are called holomorphic Volterra Mellin symbols.

Remark 5.2.2. Recall that the embedding

Lµ;ℓV (cl)(X ; Hβ;E,F ) → Lµ;ℓ

(cl)(X ; Γβ;E,F )

is well-defined and continuous in view of Proposition 2.3.2 and the considerationsin Section 3.2. Using Proposition 5.1.7 we conclude that also the embedding

Mµ;ℓV,P (cl)(X ; Hβ;E,F ) →Mµ;ℓ

P (cl)(X ;E,F )

is well-defined and continuous.Moreover, the spaces of meromorphic Volterra Mellin symbols are indepen-

dent of the right half-plane Hβ as far as Hβ ∩ πCP = ∅. This follows from (2.1.1)


together with the considerations about the translation operator in parameter-dependent Volterra pseudodifferential operators from Sections 2.3 and 3.2 (seealso Proposition 2.6.3).

In particular, holomorphic Volterra Mellin symbols are parameter-dependentVolterra pseudodifferential operators with respect to any right half-plane Hβ ⊆C. Therefore, we suppress the half-plane from the notation when we deal withholomorphic Volterra Mellin symbols.

Proposition 5.2.3. Let µ, µ′ ∈ R, µ′ ≤ µ. Moreover, let P ∈ As(L−∞(X ;E,F )

)

and β ∈ R with Hβ ∩ πCP = ∅. The following identity holds algebraically andtopologically:

Mµ;ℓP (X ;E,F ) ∩ Lµ

′;ℓV (X ; Hβ;E,F ) = Mµ′;ℓ

V,P (X ; Hβ;E,F ).

For holomorphic Volterra Mellin symbols we have:

a) The complex derivative acts continuous in the spaces

∂z : Mµ;ℓV,O(cl)(X ;E,F ) →Mµ−ℓ;ℓ

V,O(cl)(X ;E,F ).

b) For a ∈ Mµ;ℓV,O(cl)(X ;E,F ) the following asymptotic expansion holds for a|Hβ0

in terms of a|Hβ which depends smoothly on (β0, β) ∈ R × R:

a|Hβ0 ∼V

∞∑

k=0

(β0 − β)k

k!

(∂kz a

)|Hβ .

In the classical case we thus obtain for arbitrary β0, β ∈ R the following rela-tionship for the parameter-dependent homogeneous principal symbol:

σµ;ℓψ (a|Hβ0 ) = σµ;ℓ

ψ (a|Hβ ).

c) For s, ν ∈ R with ν ≥ µ we have a continuous embedding

Mµ;ℓV,O(X ;E,F ) →

Sµℓ

V,O(C;Hs(X,E), Hs−ν(X,F )) ν ≥ 0

Sµ−νℓ

V,O (C;Hs(X,E), Hs−ν(X,F )) ν ≤ 0.

Proof. These assertions follow from Proposition 5.1.7 and Remark 5.2.2. The as-ymptotic expansion in b) follows as in the proof of Lemma 5.1.2 from Proposition2.6.3. For c) see also Theorem 3.2.6.

Theorem 5.2.4. a) Let P,Q ∈ As(L−∞(X ;E,F )

)such that

(πCP ∪ πCQ

)∩

Hβ = ∅. Then pointwise addition as Lµ(X ;E,F )-valued operator functions onC \

(πCP ∪ πCQ

)induces a bilinear and continuous mapping

+ : Mµ;ℓV,P (cl)(X ; Hβ;E,F ) ×Mµ;ℓ

V,Q(cl)(X ; Hβ;E,F ) −→Mµ;ℓV,R(cl)(X ; Hβ;E,F ),

where R ∈ As(L−∞(X ;E,F )

)is (in general) determined from P and Q as in

Theorem 5.1.8.


b) Let G ∈ Vect(X) be another vector bundle, and let a ∈ Mµ;ℓV,P (cl)(X ; Hβ;G,F ),

as well as b ∈Mµ′;ℓV,Q(cl)(X ; Hβ;E,G). Then the pointwise composition (multipli-

cation) as operator functions on C\(πCP ∪πCQ

)gives rise to an element ab ∈

Mµ+µ′;ℓV,R(cl)(X ; Hβ;E,F ) with a resulting asymptotic type R ∈ As

(L−∞(X ;E,F )

)

which is determined from P and Q as in Theorem 5.1.8.c) For holomorphic Volterra Mellin symbols the multiplication as functions on C

gives rise to a continuous bilinear mapping

· : Mµ;ℓV,O(cl)(X ;G,F ) ×Mµ′;ℓ

V,O(cl)(X ;E,G) −→Mµ+µ′;ℓV,O(cl)(X ;E,F ).

Proof. These assertions follow from Theorem 5.1.8 and Section 3.2, see in partic-ular Theorem 3.2.5 what the composition is concerned.

Theorem 5.2.5. The Mellin kernel cut-off operator (cf. Remark 5.1.9, Theorem5.1.10) with respect to the weight γ ∈ R restricts to continuous bilinear mappingsin the spaces

Hγ :

C∞B (R+)×Lµ;ℓ

V (cl)(X ; H 12−γ

;E,F ) −→ Lµ;ℓV (cl)(X ; H 1

2−γ;E,F )

C∞0 (R+)×Lµ;ℓ

V (cl)(X ; H 12−γ

;E,F ) −→Mµ;ℓV,O(cl)(X ;E,F ).

Given ϕ ∈ C∞B (R+) and a ∈ Lµ;ℓ

V (X ; H 12−γ

;E,F ) we have the following asymptotic

expansion of Hγ(ϕ)a in terms of ϕ and a in the sense of Definition 3.2.7:

Hγ(ϕ)a ∼V

∞∑

k=0

1

k!(r∂r)

kϕ(r)|r=1 · ∂kz a.

If ψ ∈ C∞0 (R+) such that ψ ≡ 1 near r = 1 then the operator I − Hγ(ψ) is

continuous in the spaces

I −Hγ(ψ) : Lµ;ℓV (X ; H 1

2−γ;E,F ) −→ L−∞

V (X ; H 12−γ

;E,F ).

Proof. This follows as Remark 5.1.9, Theorem 5.1.10 analogously to Theorem 3.2.9and Corollary 3.2.10 from Theorem 2.6.13.

Corollary 5.2.6. For P ∈ As(L−∞(X ;E,F )

)such that πCP ∩ Hβ = ∅ we have

Mµ;ℓV,P (cl)(X ; Hβ;E,F ) = Mµ;ℓ

V,O(cl)(X ;E,F ) +M−∞V,P (X ; Hβ;E,F )

algebraically and topologically with the topology of the non-direct sum of Frechetspaces on the right-hand sides.

Proof. Let ψ ∈ C∞0 (R+) such that ψ ≡ 1 near r = 1. In view of Theorem 5.2.5

and Theorem 5.2.4 we may write for a ∈ Mµ;ℓV,P (cl)(X ; Hβ;E,F ) analogously to

Corollary 5.1.11a = H 1

2−β(ψ)a+

(I −H 1

2−β(ψ)

)a,

where H 12−β

(ψ)a ∈Mµ;ℓV,O(cl)(X ;E,F ), and

(I −H 1

2−β(ψ)

)a is an element of

L−∞V (X ; Hβ;E,F ) ∩Mµ;ℓ

V,P (cl)(X ; Hβ;E,F ) = M−∞V,P (X ; Hβ;E,F )


due to Proposition 5.2.3 and Theorem 5.2.4. This provides a topological isomor-phism as asserted.

Definition 5.2.7. Let P ∈ As(L−∞(X ;E,F )

)such that Hβ ∩ πCP = ∅. An

element a ∈Mµ;ℓV,P (cl)(X ; Hβ;E,F ) is called parabolic, if a|Hβ ∈ Lµ;ℓ

V (cl)(X ; Hβ;E,F )

is parabolic in the sense of Definition 3.2.17.According to Corollary 5.2.6 we may write a = a0 + r with a0 ∈

Mµ;ℓV,O(cl)(X ;E,F ) and r ∈ M−∞

V,P (X ; Hβ;E,F ). Thus we see that a is parabolic if

and only if a0 is parabolic, i. e., a|Hβ is parabolic in the sense of Definition 3.2.17.But the latter condition is independent of the particular choice of the half-planeHβ according to Proposition 5.2.3. In this sense we may speak about parabolic-ity for meromorphic Volterra Mellin symbols without referring to the particularhalf-plane involved.

Theorem 5.2.8. An element a ∈ Mµ;ℓV,P (cl)(X ; Hβ;E,F ) is parabolic if and only

if there exists some β′ ≥ β and b ∈ M−µ;ℓV,Q(cl)(X ; Hβ′;F,E) such that a·b ≡ 1 and

b·a ≡ 1, i. e., a is invertible as a meromorphic operator function with a−1 = b ∈M−µ;ℓV,Q(cl)(X ; Hβ′ ;F,E). If a(z) : Hs(X,E) → Hs−µ(X,F ) is invertible for some

s ∈ R for all z ∈ Hβ we may choose β′ = β.

Proof. Let a ∈Mµ;ℓV,P (cl)(X ; Hβ;E,F ) be parabolic. Then a is elliptic as an element

of Mµ;ℓP (cl)(X ;E,F ). Consequently we may apply Theorem 5.1.14 which shows that

a is invertible as a meromorphic operator function with a−1 = b ∈M−µ;ℓQ(cl)(X ;F,E).

For a|Hβ ∈ Lµ;ℓV (cl)(X ; Hβ;E,F ) is parabolic we may apply Theorem 3.2.19 to a|Hβ .

This shows that b necessarily belongs to the space M−µ;ℓV,Q(cl)(X ; Hβ′;F,E) with

some β′ ≥ β, where we may choose β′ = β if a(z) is pointwise invertible in theSobolev spaces for z ∈ Hβ. This proves the theorem, for the converse is immediate.

Mellin quantization.

Remark 5.2.9. For later purposes let us note, that the Mellin quantization operatorQ and its inverse Q, see Definition 2.6.15, are well-behaved within parameter-dependent Volterra operators. The proof is analogous to that of Theorem 3.2.9,and it is based on Theorem 2.6.16 in the abstract framework. Hence we restrictourselves to state the result.

Theorem 5.2.10. a) The operator Q from (2.6.6) restricts to continuous bilinearmappings

Q :

C∞

0 (R+)×Lµ;ℓ(cl)(X ; R;E,F ) −→Mµ;ℓ

O(cl)(X ;E,F )

C∞0 (R+)×Lµ;ℓ

V (cl)(X ; H;E,F ) −→Mµ;ℓV,O(cl)(X ;E,F ).

Moreover, the asymptotic expansion result (2.6.8) of Q(ϕ, a)|Γ 12−γ

, respectively

Q(ϕ, a)|H 12−γ

, in terms of a is valid in the sense of the Definitions 3.1.8 and


3.2.7:

Q(ϕ, a)(1

2− γ + iτ) ∼

(V )ϕ(1)a(−τ) +

∞∑

k=1

k∑

j=0

ck,j(ϕ, γ)(−τ)j(∂k+jτ a

)(−τ)

for τ ∈ R, respectively τ ∈ H−.

b) The operator Qγ from (2.6.7) restricts to continuous bilinear mappings

Qγ :

C∞

0 (R+)×Lµ;ℓ(cl)(X ; Γ 1

2−γ;E,F ) −→Mµ;ℓ

iO(cl)(X ;E,F )

C∞0 (R+)×Lµ;ℓ

V (cl)(X ; H 12−γ

;E,F ) −→Mµ;ℓV,iO(cl)(X ;E,F ).

The spaces in the image are the multiples by the imaginary unit in the parameterof the ordinary spaces of meromorphic Mellin symbols.

The asymptotic expansion (2.6.9) of Qγ(ψ, a)|R, respectively Qγ(ψ, a)|H, interms of a is valid in the sense of the Definitions 3.1.8 and 3.2.7:

Qγ(ψ, a)(τ) ∼(V )

ψ(1)a(1

2− γ − iτ) +

∞∑

k=1

k∑

j=0

dk,j(ψ, γ)(−iτ)j(∂k+jτ a

)(1

2− γ − iτ)

for τ ∈ R, respectively τ ∈ H.c) For ϕ, ψ ∈ C∞

0 (R+) such that ϕ ≡ 1 and ψ ≡ 1 near r = 1 we have

Q(Qγ(a)) − a ∈L−∞(X ; Γ 1

2−γ;E,F )

L−∞V (X ; H 1

2−γ;E,F ),

Qγ(Q(a)) − a ∈L−∞(X ; R;E,F )

L−∞V (X ; H;E,F ).

5.3. Elements of the Mellin calculus

Remark 5.3.1. In this section we introduce subcalculi of the Mellin pseudodiffer-ential calculi with operator-valued symbols from Sections 2.5 and 2.6, where thesymbols are built upon parameter-dependent pseudodifferential operators on themanifold X . To this end recall from Theorem 3.1.5, Theorem 3.2.6, Proposition5.1.7 and Proposition 5.2.3 the following embeddings:

Lµ;ℓ(X ; Γβ;E,F ) → Sµℓ (Γβ ;H

s(X,E), Hs−µ+(X,F )),

Lµ;ℓV (X ; Hβ;E,F ) → S

µℓ

V (Hβ;Hs(X,E), Hs−µ+(X,F )),

Mµ;ℓO (X ;E,F ) → S

µℓ

O (C;Hs(X,E), Hs−µ+(X,F )),

Mµ;ℓV,O(X ;E,F ) → S

µℓ

V,O(C;Hs(X,E), Hs−µ+(X,F )).

for s, µ ∈ R, where µ+ := max0, µ.


In particular, for every s ∈ R we have

C∞B ((R+)q, Lµ;ℓ(X ;Γ 1

2−γ;E,F )) →

MγSµℓ ((R+)q × Γ 1

2−γ;Hs(X,E), Hs−µ+(X,F )),

C∞B ((R+)q, Lµ;ℓ

V (X ;H 12−γ

;E,F )) →

MγSµℓ

V ((R+)q × H 12−γ

;Hs(X,E), Hs−µ+(X,F )),

for q = 1, 2, see Definition 2.5.1. This shows that for every double-symbola ∈ C∞

B (R+ × R+, Lµ;ℓ(X ; Γ 1

2−γ;E,F )) the associated Mellin pseudodifferential

operator acts continuously in the spaces

opγM (a) : Tγ(X∧, E) −→ Tγ(X∧, F ),

and left- or right-symbols a are uniquely determined by this action in view of The-orem 2.5.4 and the density of Tγ(X∧, E) in Tγ(R+, H

t(X,E)) for every t ∈ R. Asturns out, the classes of Mellin pseudodifferential operators based on such symbolsremain preserved by the manipulations in the (holomorphic) Mellin calculus fromSection 2.5 and Section 2.6.

Theorem 5.3.2. Consider a double-symbol a belonging to one of the followingspaces:

i) C∞B (R+ × R+, L

µ;ℓ(cl)(X ; Γ 1

2−γ;E,F )),

ii) C∞B (R+ × R+, L

µ;ℓV (cl)(X ; H 1

2−γ;E,F )),

iii) C∞B (R+ × R+,M

µ;ℓO(cl)(X ;E,F )),

iv) C∞B (R+ × R+,M

µ;ℓV,O(cl)(X ;E,F )).

Then the corresponding left- and right-symbols aL and aR obtained from Theorem2.5.4 in the cases i) and ii), respectively from Theorem 2.6.7 in the cases iii) andiv), belong to the spaces

i) C∞B (R+, L

µ;ℓ(cl)(X ; Γ 1

2−γ;E,F )),

ii) C∞B (R+, L


2−γ;E,F )),

iii) C∞B (R+,M


iv) C∞B (R+,M

µ;ℓV,O(cl)(X ;E,F )).

Moreover, the asymptotic expansions of aL and aR in terms of a from Theorem2.5.4 and 2.6.7 are valid within these smaller classes (see also the Definitions 3.1.8and 3.2.7).

Proof. From the correspondence (3.1.4) and (3.1.3) we conclude that the proofof the theorem reduces to consider local symbols and global remainders of order−∞, since we have explicit oscillatory integral formulas for the left- and right-symbol at hand. Keeping in mind the characterization of the remainders on themanifold as operator-valued symbols (cf. Definition 3.1.2 and 3.2.2) we see that thecases i) and ii) follow from Theorem 2.5.4, while iii) and iv) follow from Theorem


2.6.7. Note that the global asymptotic expansions on the manifold follow fromthe corresponding asymptotic expansions on the level of local symbols, which aresubject to the theorems in the abstract framework.

Theorem 5.3.3. a) Let a and b be given, where

i) a ∈ C∞B (R+, L

µ;ℓ(cl)(X ; Γ 1

2−γ;F,G)), b ∈ C∞

B (R+, Lµ′;ℓ(cl) (X ; Γ 1

2−γ;E,F )),

ii) a ∈ C∞B (R+, L


2−γ;F,G)), b ∈ C∞

B (R+, Lµ′;ℓV (cl)(X ; H 1

2−γ;E,F )),

iii) a ∈ C∞B (R+,M

µ;ℓO(cl)(X ;F,G)), b ∈ C∞

B (R+,Mµ′;ℓO(cl)(X ;E,F )),

iv) a ∈ C∞B (R+,M

µ;ℓV,O(cl)(X ;F,G)), b ∈ C∞

B (R+,Mµ′;ℓV,O(cl)(X ;E,F )).

Then the Leibniz-product a#b (cf. Theorem 2.5.6 and Theorem 2.6.9) belongsto

i) C∞B (R+, L

µ+µ′;ℓ(cl) (X ; Γ 1

2−γ;E,G)),

ii) C∞B (R+, L

µ+µ′;ℓV (cl) (X ; H 1

2−γ;E,G)),

iii) C∞B (R+,M

µ+µ′;ℓO(cl) (X ;E,G)),

iv) C∞B (R+,M

µ+µ′;ℓV,O(cl)(X ;E,G)),

and the asymptotic expansions (2.5.3), (2.5.4) and (2.6.1) hold within thesmaller classes. The formulas for the conormal symbols of the composition inthe cases iii) and iv) are inherited from the abstract framework; see Definition2.6.10, in particular the defining relation (2.6.3) and (2.6.4). In the classicalcase we conclude that the homogeneous principal symbol of the Leibniz-productis given as the product of the homogeneous principal symbols of a and b.

b) Let

i) a ∈ C∞B (R+, L

µ;ℓ(cl)(X ; Γn+1

2 −γ ;E,F )),

ii) a ∈ C∞B (R+,M


and A = opγ−n

2

M (a). Then the formal adjoint with respect to the r−n2 L2-inner

product is given as A∗ = op−γ−n

2

M (a∗) with the symbol

a∗(r, z) =(a(r′, n+ 1 − z)(∗)

)L.

Here (∗) denotes the formal adjoint with respect to the L2-inner product on themanifold. This shows that

i) a∗ ∈ C∞B (R+, L

µ;ℓ(cl)(X ; Γn+1

2 +γ ;F,E)),

ii) a∗ ∈ C∞B (R+,M

µ;ℓO(cl)(X ;F,E)).

Moreover, the following asymptotic expansion of a∗ in terms of a is valid:

a∗(r,n+ 1

2+ γ + iτ

)∼

∞∑

k=0

1

k!(−1)kDk

τ (−r∂r)ka(r,n+ 1

2− γ + iτ

)(∗).

In the classical case we thus obtain the following formula for the (anisotropic)homogeneous principal symbol:

σµ;ℓψ (A∗) = σµ;ℓ

ψ (A)∗.


In case of ii) we have the following relations for the conormal symbols:

σ−kM

(A∗

)(z) = σ−k

M

(A

)(n+ 1 − k − z)(∗)

for k ∈ N0.

Proposition 5.3.4. Let a belong to one of the following spaces

i) C∞B (R+ × R+, L

µ;ℓ(X ; Γ 12−γ

;E,F )),

ii) C∞B (R+ × R+, L

µ;ℓV (X ; H 1

2−γ;E,F )),

iii) C∞B (R+ × R+,M

µ;ℓO (X ;E,F )),

iv) C∞B (R+ × R+,M

µ;ℓV,O(X ;E,F )),

and assume that a(r, r′) ≡ 0 for∣∣ rr′ − 1

∣∣ < ε for a sufficiently small ε > 0. ThenopγM (a) = opγM (c) with a symbol c in

i) C∞B (R+, L

−∞(X ; Γ 12−γ

;E,F )),

ii) C∞B (R+, L

−∞V (X ; H 1

2−γ;E,F )),

iii) C∞B (R+,M

−∞O (X ;E,F )),

iv) C∞B (R+,M

−∞V,O (X ;E,F )).

Proposition 5.3.5. Let a ∈ C∞B (R+, L

µ;ℓV (X ; H 1

2−γ;E,F )). Then opγM (a) re-

stricts for every r0 ∈ R+ to a continuous operator

opγM (a) : Tγ,0((0, r0), C∞(X,E)) −→ Tγ,0((0, r0), C∞(X,F )).

Proof. This follows from Theorem 2.5.8.

Theorem 5.3.6. Let a ∈ C∞B (R+, L

µ;ℓ(X ; Γn+12 −γ ;E,F )). Then op

γ−n2

M (a) ex-

tends for every s, t ∈ R to a continuous operator

opγ−n

2

M (a) : H(s,t),γ;ℓ(X∧, E) −→ H(s−µ,t),γ;ℓ(X∧, F ).

If a ∈ C∞B (R+, L

µ;ℓV (X ; Hn+1

2 −γ ;E,F )), then opγ−n

2

M (a) restricts for every r0 ∈ R+

to a continuous operator

opγ−n

2

M (a) : H(s,t),γ;ℓ0 ((0, r0]×X,E) −→ H(s−µ,t),γ;ℓ

0 ((0, r0]×X,F ).

Proof. Let

R−s(z) ∈ L−s;ℓ(X ; Γn+12 −γ ;E,E),

Rs−µ(z) ∈ Ls−µ;ℓ(X ; Γn+12 −γ ;F, F ),

be parameter-dependent reductions of orders according to Theorem 3.1.12. In viewof Definition 4.2.3 the asserted boundedness is equivalent to the continuity of

opγ−n

2

M (Rs−µ#a#R−s) : H(0,t),γ;ℓ(X∧, E) −→ H(0,t),γ;ℓ(X∧, F ).

Recall that H(0,t),γ;ℓ(X∧, E) = L2,γ−n2 (R+, H

t(X,E)), and that Rs−µ#a#R−s

belongs to

C∞B (R+, L

0;ℓ(X ; Γn+12 −γ ;E,F )) → Mγ−n

2S0(R+×Γn+1

2 −γ ;Ht(X,E), Ht(X,F )).


Consequently, we obtain the desired boundedness from Theorem 2.5.11. The secondassertion follows from Proposition 5.3.5 and Proposition 4.2.12.

Ellipticity and Parabolicity.

Remark 5.3.7. Let

a ∈C∞B (R+, L

µ;ℓ(cl)(X ; Γ 1

2−γ;E,F ))

C∞B (R+, L


2−γ;E,F )).

According to Remark 3.1.4, 3.2.4 we associate to a a complete symbol (a1, . . . , aN)subordinate to the given covering of X from Notation 3.1.1. Thus we have

(a1, . . . , aN ) ∈

C∞B (R+,

N×j=1

Sµ;ℓ(cl)(R

n × Rn × Γ 12−γ

; CN− ,CN+))

C∞B (R+,

N×j=1

Sµ;ℓV (cl)(R

n × Rn × H 12−γ

; CN− ,CN+)).

Definition 5.3.8. a) Let a ∈ C∞B (R+, L

µ;ℓ(cl)(X ; Γ 1

2−γ;E,F )). Let either I = R+

or I = (0, r0] with r0 ∈ R+. We call a elliptic (on I), if for the complete symbol(a1, . . . , aN ) we have:

For every j = 1, . . . , N there exists R > 0 and a neighbourhoodU(κj(suppψj)) such that for (ξ, τ) ∈ Rn × R with |(ξ, τ)|ℓ ≥ R and all r ∈ Iand x ∈ U(κj(suppψj)) there exists aj(r, x, ξ,

12 − γ + iτ)−1 with

‖aj(r, x, ξ,

1

2− γ + iτ

)−1‖ = O(〈ξ, τ〉−µℓ

)

as |(ξ, τ)|ℓ → ∞, uniformly for r ∈ I and x ∈ U(κj(suppψj)).In the classical case this condition is equivalent to require that the homoge-

neous principal symbol

σµ;ℓψ (a) ∈ C∞

B (R+, S(µ;ℓ)((T ∗X × Γ 1

2−γ) \ 0,Hom(π∗E, π∗F )))

is invertible (on I), and for the inverse we have

sup‖σµ;ℓψ (a)(r, ξx, τ)

−1‖; r ∈ I,(|ξx|2ℓx + |τ |2

) 12ℓ= 1 <∞.

Note that we identified Γ 12−γ

with R via τ = Im(z).

b) Let a ∈ C∞B (R+,M

µ;ℓO(cl)(X ;E,F )). Let either I = R+ or I = [0, r0] with

r0 ∈ R+. We call a elliptic (on I), if there exists γ ∈ R such that the conditionsin a) hold with the interval I.

c) Let a ∈ C∞B (R+, L


2−γ;E,F )). Let either I = R+ or I = (0, r0] with

r0 ∈ R+. We call a parabolic (on I) if the following condition is fulfilled:For every j = 1, . . . , N there exists R > 0 and a neighbourhood

U(κj(suppψj)) such that for (ξ, ζ) ∈ Rn × H0 with |(ξ, ζ)|ℓ ≥ R and all r ∈ Iand x ∈ U(κj(suppψj)) there exists aj(r, x, ξ,

12 − γ + ζ)−1 with

‖aj(r, x, ξ,

1

2− γ + ζ

)−1‖ = O(〈ξ, ζ〉−µℓ

)


as |(ξ, ζ)|ℓ → ∞, uniformly for r ∈ I and x ∈ U(κj(suppψj)).In the classical case this condition is equivalent to require that the homoge-

neous principal symbol

σµ;ℓψ (a) ∈ C∞

B (R+, S(µ;ℓ)V ((T ∗X × H 1

2−γ) \ 0,Hom(π∗E, π∗F )))


sup‖σµ;ℓψ (a)(r, ξx, ζ)

−1‖; r ∈ I,(|ξx|2ℓx + |ζ|2

) 12ℓ= 1 <∞.

Here we identified H 12−γ

with H0 via translation.

d) Let a ∈ C∞B (R+,M

µ;ℓV,O(cl)(X ;E,F )). Let either I = R+ or I = [0, r0] with

r0 ∈ R+. We call a parabolic (on I), if there exists γ ∈ R such that theconditions in c) hold with the interval I.

If I = R+ in a), c) or I = R+ in b), d) we say that a is elliptic, respectivelyparabolic, without refering to the interval.

Lemma 5.3.9. a) Let a ∈ C∞B (R+, L

µ;ℓ(cl)(X ; Γ 1

2−γ;E,F )) and I = R+ or I =

(0, r0] with r0 ∈ R+. Then the following are equivalent:• a is elliptic (on I).

• There exists b ∈ C∞B (R+, L

−µ;ℓ(cl) (X ; Γ 1

2−γ;F,E)) such that ab − 1 as

well as ba − 1 coincide in a neighbourhood of I with symbols belonging

to C∞B (R+, L

−1;ℓ(cl) (X ; Γ 1

2−γ;F, F )) and C∞

B (R+, L−1;ℓ(cl) (X ; Γ 1

2−γ;E,E)), re-

spectively.


µ;ℓO(cl)(X ;E,F )) and I = R+ or I = [0, r0] with r0 ∈ R+.

Then the following are equivalent:• a is elliptic (on I).

• There exists b ∈ C∞B (R+,M

−µ;ℓO(cl)(X ;F,E)) such that ab − 1 and ba − 1

coincide in a neighbourhood of I with symbols in C∞B (R+,M

−1;ℓO(cl)(X ;F, F ))

and C∞B (R+,M

−1;ℓO(cl)(X ;E,E)), respectively.

c) Let a ∈ C∞B (R+, L


2−γ;E,F )) and I = R+ or I = (0, r0] with r0 ∈

R+. Then the following are equivalent:• a is parabolic (on I).

• There exists b ∈ C∞B (R+, L

−µ;ℓV (cl)(X ; H 1

2−γ;F,E)) such that ab − 1 and

ba − 1 coincide in a neighbourhood of I with symbols belonging to

C∞B (R+, L

−1;ℓV (cl)(X ; H 1

2−γ;F, F )) and C∞

B (R+, L−1;ℓV (cl)(X ; H 1

2−γ;E,E)), re-

spectively.

d) Let a ∈ C∞B (R+,M

µ;ℓV,O(cl)(X ;E,F )) and I = R+ or I = [0, r0] with r0 ∈ R+.

Then the following are equivalent:• a is parabolic (on I).


−µ;ℓV,O(cl)(X ;F,E)) such that ab−1 and ba−1 co-

incide in a neighbourhood of I with symbols in C∞B (R+,M

−1;ℓV,O(cl)(X ;F, F ))

and C∞B (R+,M

−1;ℓV,O(cl)(X ;E,E)), respectively.


In particular, the notions of ellipticity and parabolicity on an interval I from Def-inition 5.3.8 are well-defined, i. e., independent of the choice of the data on Xand the subordinated complete symbol, as well as independent of the choice of theparticular weight line or right half-plane for analytic symbols.

Proof. a) follows from Theorem 3.1.10 and c) follows from Theorem 3.2.18. For theproof of b) and d) note first that the existence of symbols b with the asserted prop-erties is sufficient for the ellipticity or parabolicity of a on the interval I in view of

Theorem 3.1.10 and Theorem 3.2.18. Now let a ∈ C∞B (R+,M

µ;ℓV,O(cl)(X ;E,F ))

be parabolic on the interval I. From Theorem 3.2.18 we obtain a symbol

b ∈ C∞B (R+, L

−µ;ℓV (cl)(X ; H 1

2−γ;F,E)) such that ab − 1 and ba − 1 coincide in a

neighbourhood of I with symbols belonging to C∞B (R+, L

−1;ℓV (cl)(X ; H 1

2−γ;F, F ))

and C∞B (R+, L

−1;ℓV (cl)(X ; H 1

2−γ;E,E)), respectively, for some γ ∈ R. Let ϕ ∈

C∞0 (R+) such that ϕ ≡ 1 near r = 1, and define b := Hγ(ϕ)b with the

Mellin kernel cut-off operator Hγ . Then we obtain from Theorem 5.2.5 that b ∈C∞B (R+,M

−µ;ℓV,O(cl)(X ;F,E)), and we have b− b ∈ C∞

B (R+, L−∞V (cl)(X ; H 1

2−γ;F,E)).

Consequently, ab− 1 and ba− 1 coincide in a neighbourhood of I with symbols in

C∞B (R+, L

−1;ℓV (cl)(X ; H 1

2−γ;F, F )) respectively C∞

B (R+, L−1;ℓV (cl)(X ; H 1

2−γ;E,E)), but

both ab−1 and ba−1 are analytic symbols. Thus we obtain from Proposition 5.2.3the desired assertion which completes the proof of d). The proof of b) is analogous.

Theorem 5.3.10. a) Let a ∈ C∞B (R+, L


2−γ;E,F )). The following are

equivalent:• a is parabolic.

• There exists a symbol b ∈ C∞B (R+, L

−µ;ℓV (cl)(X ; H 1

2−γ;F,E)) such that

opγM (a)opγM (b) = 1 + opγM (rR),

opγM (b)opγM (a) = 1 + opγM (rL),

with remainders

rR ∈ C∞B (R+, L

−∞V (X ; H 1

2−γ;F, F )),

rL ∈ C∞B (R+, L

−∞V (X ; H 1

2−γ;E,E)).

Let I = (0, r0] for some r0 ∈ R+. Then the following are equivalent:• a is parabolic on I.


−µ;ℓV (cl)(X ; H 1

2−γ;F,E)) as well as cut-

off functions ω, ω ∈ C∞0 (R+) near r = 0 with ω, ω ≡ 1 on I, such that

ω(opγM (a)opγM (b) − 1

)ω = opγM (rR),

ω(opγM (b)opγM (a) − 1

)ω = opγM (rL),


with remainders

rR ∈ C∞B (R+, L

−∞V (X ; H 1

2−γ;F, F )),

rL ∈ C∞B (R+, L

−∞V (X ; H 1

2−γ;E,E)).


µ;ℓV,O(cl)(X ;E,F )). The following are equivalent:

• a is parabolic.


−µ;ℓV,O(cl)(X ;F,E)) such that



for some (every) γ ∈ R, where

rR ∈ C∞B (R+,M

−∞V,O (X ;F, F )),

rL ∈ C∞B (R+,M

−∞V,O (X ;E,E)).

Let I = [0, r0] for some r0 ∈ R+. Then the following are equivalent:• a is parabolic on I.


−µ;ℓV,O(cl)(X ;F,E)) as well as cut-off functions

ω, ω ∈ C∞0 (R+) near r = 0 with ω, ω ≡ 1 on I such that


)ω = opγM (rR),


)ω = opγM (rL),

for some (every) γ ∈ R, where

rR ∈ C∞B (R+,M

−∞V,O (X ;F, F )),

rL ∈ C∞B (R+,M

−∞V,O (X ;E,E)).

Proof. In view of Theorem 5.3.2, Theorem 5.3.3 and Lemma 5.3.9 the above con-ditions in a) and b) are clearly sufficient for the parabolicity of the symbols (onthe interval I). Now assume that

a ∈C∞B (R+, L


2−γ;E,F ))

C∞B (R+,M

µ;ℓV,O(cl)(X ;E,F ))

is parabolic. From Lemma 5.3.9 and Theorem 5.3.3 we conclude that there exists

b ∈C∞B (R+, L

−µ;ℓV (cl)(X ; H 1

2−γ;F,E))

C∞B (R+,M

−µ;ℓV,O(cl)(X ;F,E))


such that a#b = 1 − rR and b#a = 1 − rL with

rR ∈C∞B (R+, L

−1;ℓV (cl)(X ; H 1

2−γ;F, F ))

C∞B (R+,M

−1;ℓV,O(cl)(X ;F, F )),

rL ∈C∞B (R+, L

−1;ℓV (cl)(X ; H 1

2−γ;E,E))

C∞B (R+,M

−1;ℓV,O(cl)(X ;E,E)).

Now choose rR, rL such that

C∞B (R+, L

0;ℓV (cl)(X ; H 1

2−γ;F, F ))

C∞B (R+,M

0;ℓV,O(cl)(X ;F, F ))

∋ rR ∼

V

∞∑

j=0

#(j)rR,

C∞B (R+, L

0;ℓV (cl)(X ; H 1

2−γ;E,E))

C∞B (R+,M

0;ℓV,O(cl)(X ;E,E))

∋ rL ∼

V

∞∑

j=0

#(j)rL.

These asymptotic expansions are to be carried out within the corresponding sym-bol classes. Recall that the terms in the asymptotic expansions are well-defined inthe corresponding symbol classes with decreasing orders by Theorem 5.3.3, andthat the classes themselves are closed with respect to taking asymptotic sums byTheorem 2.6.14, Theorem 3.2.12, Theorem 5.2.5 and Proposition 5.2.3. Now wesee that

a#(b#rR

)− 1 ∈

C∞B (R+, L

−∞V (X ; H 1

2−γ;F, F ))

C∞B (R+,M

−∞V,O (X ;F, F )),

(rL#b

)#a− 1 ∈

C∞B (R+, L

−∞V (X ; H 1

2−γ;E,E))

C∞B (R+,M

−∞V,O (X ;E,E)),

and consequently the same relations hold with either b := b#rR or b := rL#b.This completes the proof of the first assertions in a) and b).

Now we consider the case of a finite interval I. Let a be parabolic on I.Choose b satisfying the conditions in Lemma 5.3.9, and let rL and rR be (classical

holomorphic) Volterra symbols of order −1, such that a#b = 1 + rR and b#a =1+ rL in a neighbourhood of I. Observe that 1+ rR and 1+ rL are parabolic, andconsequently we obtain from the already proven part of the theorem that thereexist (classical holomorphic) Volterra symbols rL and rR of order −1, such that(1+ rL

)#

(1+ rL

)−1 and

(1+ rR

)#

(1+ rR

)−1 are of order −∞. Now we see that

if we set either b :=(1 + rL

)#b or b := b#

(1 + rR

), and choose cut-off functions

ω, ω ∈ C∞0 (R+) supported sufficiently close to the interval I, we obtain the second

assertions in a) and b). This finishes the proof of the theorem.

Theorem 5.3.11. a) Let a ∈ C∞B (R+, L

µ;ℓ(cl)(X ; Γ 1

2−γ;E,F )). The following are

equivalent:• a is elliptic.



−µ;ℓ(cl) (X ; Γ 1

2−γ;F,E)), and symbols

rR ∈ C∞B (R+, L

−∞(X ; Γ 12−γ

;F, F )), rL ∈ C∞B (R+, L

−∞(X ; Γ 12−γ

;E,E)),

such that


opγM (b)opγM (a) = 1 + opγM (rL).

Let I = (0, r0] for some r0 ∈ R+. Then the following are equivalent:• a is elliptic on I.


−µ;ℓ(cl) (X ; Γ 1

2−γ;F,E)), and symbols

rR ∈ C∞B (R+, L

−∞(X ; Γ 12−γ

;F, F )), rL ∈ C∞B (R+, L

−∞(X ; Γ 12−γ

;E,E)),

as well as cut-off functions ω, ω ∈ C∞0 (R+) near r = 0 with ω, ω ≡ 1 on

I, such that


)ω = opγM (rR),


)ω = opγM (rL).


µ;ℓO(cl)(X ;E,F )). The following are equivalent:

• a is elliptic.

• There exists a symbol b ∈ C∞B (R+,M

−µ;ℓO(cl)(X ;F,E)) and remainders rR ∈

C∞B (R+,M

−∞O (X ;F, F )), rL ∈ C∞

B (R+,M−∞O (X ;E,E)), such that



for some (every) γ ∈ R.Let I = [0, r0] for some r0 ∈ R+. Then the following are equivalent:

• a is elliptic on I.

• There exists a symbol b ∈ C∞B (R+,M

−µ;ℓO(cl)(X ;F,E)) and remainders rR ∈

C∞B (R+,M

−∞O (X ;F, F )), rL ∈ C∞

B (R+,M−∞O (X ;E,E)), as well as cut-

off functions ω, ω ∈ C∞0 (R+) near r = 0 with ω, ω ≡ 1 on I, such that


)ω = opγM (rR),


)ω = opγM (rL),

for some (every) γ ∈ R.

Proof. The proof is analogous to that of Theorem 5.3.10.

5.4. Elements of the Fourier calculus with global weights

Remark 5.4.1. Analogously to Section 5.3 we are going to introduce subcalculiof the pseudodifferential calculi with operator-valued symbols satisfying globalweight conditions from Section 2.7, where the symbols are built upon parameter-dependent pseudodifferential operators on the manifold X .


Recall that for every s ∈ R we have

S1,2(R2, Lµ;ℓ(X ; R;E,F )) → Sµℓ,1,2(R2 × R;Hs(X,E), Hs−µ+(X,F )),

S(R, Lµ;ℓ(X ; R;E,F )) → Sµℓ,(R × R;Hs(X,E), Hs−µ+(X,F )),

S1,2(R2, Lµ;ℓV (X ; H;E,F )) → S

µℓ,1,2

V (R2 × H;Hs(X,E), Hs−µ+(X,F )),

S(R, Lµ;ℓV (X ; H;E,F )) → S

µℓ,

V (R × H;Hs(X,E), Hs−µ+(X,F )),

where µ+ := max0, µ, see Definition 2.7.2. Consequently, for every double-symbol a ∈ S1,2(R2, Lµ;ℓ(X ; R;E,F )) the associated pseudodifferential operatoracts continuously in the spaces

opr(a) : S(R ×X,E) −→ S(R ×X,F ),

and left- or right-symbols a are uniquely determined by this action in view ofTheorem 2.7.4 and the density of S(R ×X,E) in S(R, Ht(X,E)) for every t ∈ R.The classes of pseudodifferential operators with global weight conditions based onsuch symbols are invariant with respect to the manipulations in the calculus fromSection 2.7. The technique to see this is the same as before. Therefore, we willonly state the results what the basic elements of the calculus are concerned, andskip the proofs.

Remark 5.4.2. If explicitly stated, the asymptotic expansions in the sequel are tobe regarded as follows:

Let (µk), (k) ⊆ R be sequences such that µk, k −→k→∞

−∞, and µ := maxk∈N

µk

as well as := maxk∈N

k. Moreover, let

ak ∈Sk(R, Lµk;ℓ(X ; R;E,F ))

Sk(R, Lµk;ℓV (X ; H;E,F )),

a ∈S(R, Lµ;ℓ(X ; R;E,F ))

S(R, Lµ;ℓV (X ; H;E,F )).

We write a ∼(V )

∞∑j=1

aj if for every R ∈ R there is a k0 ∈ N such that for k > k0

a−k∑

j=1

aj ∈SR(R, LR;ℓ(X ; R;E,F ))

SR(R, LR;ℓV (X ; H;E,F )).

Theorem 5.4.3. Consider a double-symbol

a ∈S1,2(R2, Lµ;ℓ

(cl)(X ; R;E,F ))

S1,2(R2, Lµ;ℓV (cl)(X ; H;E,F )).


Then the corresponding left- and right-symbols aL and aR obtained from Theorem2.7.4 belong to the spaces

aL, aR ∈S1+2(R, Lµ;ℓ

(cl)(X ; R;E,F ))

S1+2(R, Lµ;ℓV (cl)(X ; H;E,F )),

and the mappings a 7−→ aL, aR are continuous within

S1,2(R2, Lµ;ℓ(cl)(X ; R;E,F ))

S1,2(R2, Lµ;ℓV (cl)(X ; H;E,F ))

−→

S1+2(R, Lµ;ℓ

(cl)(X ; R;E,F ))

S1+2(R, Lµ;ℓV (cl)(X ; H;E,F )).

Moreover, the asymptotic expansions of aL and aR in terms of a from Theorem2.7.4 are valid in the sense of Remark 5.4.2.

Theorem 5.4.4. a) Let

a ∈S(R, Lµ;ℓ

(cl)(X ; R;F,G))

S(R, Lµ;ℓV (cl)(X ; H;F,G))

and b ∈S

′

(R, Lµ′;ℓ

(cl) (X ; R;E,F ))

S′

(R, Lµ′;ℓV (cl)(X ; H;E,F )).

Then we have for the Leibniz-product (cf. Theorem 2.7.5)

a#b ∈S+

′

(R, Lµ+µ′;ℓ(cl) (X ; R;E,G))

S+′

(R, Lµ+µ′;ℓV (cl) (X ; H;E,F )).

Moreover, the asymptotic expansion

a#b ∼(V )

∞∑

k=0

1

k!(∂kτ a)(D

kr b)

holds in the sense of Remark 5.4.2.b) Let [·] : R → R+ be a smoothed norm function, and δ ∈ R. Moreover, let

a ∈ S(R, Lµ;ℓ(cl)(X ; R;E,F )). Let a(∗),δ be the formal adjoint symbol with respect

to the [·]−δL2-inner product (see also Theorem 2.7.9). Then we have

a(∗),δ =([r]−2δ

(a(r′, τ)

)(∗)[r′]2δ

)L∈ S(R, Lµ;ℓ

(cl)(X ; R;F,E)),

and we have the asymptotic expansion

a(∗),δ(r, τ) ∼∞∑

k=0

∑

p+q=k

1

p!q!

([r]−2δDp

r [r]2δ

)(∂kτD

qr(a(r, τ))

(∗))

in the sense of Remark 5.4.2. Here (∗) denotes the formal adjoint with respectto the L2-inner product on the manifold.

Proposition 5.4.5. Let

a ∈S1,2(R2, Lµ;ℓ(X ; R;E,F ))

S1,2(R2, Lµ;ℓV (X ; H;E,F ))


such that a(r, r′) ≡ 0 for∣∣r− r′

∣∣ < ε for a sufficiently small ε > 0. Then opr(a) =opr(c) with a symbol

c ∈S−∞(R, L−∞(X ; R;E,F ))

S−∞(R, L−∞V (X ; H;E,F )).

Proposition 5.4.6. Let a ∈ S(R, Lµ;ℓV (X ; H;E,F )). Then opr(a) restricts for

every r0 ∈ R+ to a continuous operator

opr(a) : S((−∞, r0), C∞(X,E)) −→ S((−∞, r0), C

∞(X,F )).

Theorem 5.4.7. Let a ∈ S(R, Lµ;ℓ(X ; R;E,F )).Then opr(a) extends for everys, t, δ ∈ R to a continuous operator

opr(a) : H(s,t);ℓ(R ×X,E)δ −→ H(s−µ,t);ℓ(R ×X,F )δ−.

If a ∈ S(R, Lµ;ℓV (X ; H;E,F )), then opr(a) restricts for every r0 ∈ R+ to a con-

tinuous operator

opr(a) : H(s,t);ℓ0 ((−∞, r0] ×X,E)δ −→ H

(s−µ,t);ℓ0 ((−∞, r0] ×X,F )δ−.

Ellipticity and Parabolicity.

Remark 5.4.8. Let

a ∈S(R, Lµ;ℓ

(cl)(X ; R;E,F ))

S(R, Lµ;ℓV (cl)(X ; H;E,F )).

As in Remark 5.3.7 we associate to a a complete symbol (a1, . . . , aN ) subordinateto the given covering of X from Notation 3.1.1. Then we have

(a1, . . . , aN ) ∈

S(R,N×j=1

Sµ;ℓ(cl)(R

n × Rn × R; CN− ,CN+))

S(R,N×j=1

Sµ;ℓV (cl)(R

n × Rn × H; CN− ,CN+)).

Definition 5.4.9. a) Let a ∈ S(R, Lµ;ℓ(cl)(X ; R;E,F )). Let either I = R or I =

[r0,∞), I = (−∞, r0] with r0 ∈ R. We call a interior elliptic (on I), if for thecomplete symbol (a1, . . . , aN ) we have:

For every j = 1, . . . , N there exists R > 0 and a neighbourhoodU(κj(suppψj)) such that for (ξ, τ) ∈ Rn × R with |(ξ, τ)|ℓ ≥ R and all r ∈ Iand x ∈ U(κj(suppψj)) there exists aj(r, x, ξ, τ)

−1 with

sup‖aj(r, x, ξ, τ

)−1‖ 〈ξ, τ〉µℓ 〈r〉; |(ξ, τ)|ℓ ≥ R, r ∈ I, x ∈ U(κj(suppψj))<∞.

In the classical case this condition is equivalent to require that the homoge-neous principal symbol

σµ;ℓψ (a) ∈ S(R, S(µ;ℓ)((T ∗X × R) \ 0,Hom(π∗E, π∗F )))



sup‖σµ;ℓψ (a)(r, ξx, τ)

−1‖〈r〉; r ∈ I,(|ξx|2ℓx + |τ |2

) 12ℓ= 1 <∞.

We call a elliptic (on I), if a is interior elliptic (on I), and there exists somes0 ∈ R such that

a(r, τ) : Hs0(X,E) −→ Hs0−µ(X,F )

is invertible for all τ ∈ R and |r| sufficiently large (on I), and

sup‖a(r, τ)−1‖L(Hs0−µ,Hs0 )〈r〉〈τ〉M ; |r| > R, τ ∈ R <∞for some R,M ∈ R.

b) Let a ∈ S(R, Lµ;ℓV (cl)(X ; H;E,F )). Let either I = R or I = [r0,∞), (−∞, r0]

with r0 ∈ R. We call a interior parabolic (on I), if the following condition isfulfilled:

For every j = 1, . . . , N there exists R > 0 and a neighbourhoodU(κj(suppψj)) such that for (ξ, ζ) ∈ Rn × H with |(ξ, ζ)|ℓ ≥ R and all r ∈ Iand x ∈ U(κj(suppψj)) there exists aj(r, x, ξ, ζ)

−1 with

sup‖aj(r, x, ξ, ζ

)−1‖ 〈ξ, ζ〉µℓ 〈r〉; |(ξ, ζ)|ℓ ≥ R, r ∈ I, x ∈ U(κj(suppψj))<∞.

In the classical case this condition is equivalent to require that the homoge-neous principal symbol

σµ;ℓψ (a) ∈ S(R, S

(µ;ℓ)V ((T ∗X × H) \ 0,Hom(π∗E, π∗F )))


sup‖σµ;ℓψ (a)(r, ξx, ζ)

−1‖〈r〉; r ∈ I,(|ξx|2ℓx + |ζ|2

) 12ℓ= 1 <∞.

We call a parabolic (on I), if a is interior parabolic (on I), and there existssome s0 ∈ R such that

a(r, ζ) : Hs0(X,E) −→ Hs0−µ(X,F )

is invertible for all ζ ∈ H and |r| sufficiently large (on I), and

sup‖a(r, ζ)−1‖L(Hs0−µ,Hs0 )〈r〉〈ζ〉M ; |r| > R, ζ ∈ H <∞for some R,M ∈ R.

If I = R in a) or b) we say that a is (interior) elliptic, respectively (interior)parabolic, without refering to the interval.

Lemma 5.4.10. a) Let a ∈ S(R, Lµ;ℓ(cl)(X ; R;E,F )). Let either I = R or I =

[r0,∞), I = (−∞, r0] with r0 ∈ R. Then the following are equivalent:• a is interior elliptic (on I).

• There exists b ∈ S−(R, L−µ;ℓ(cl) (X ; R;F,E)) such that ab − 1 and ba − 1

coincide in a neighbourhood of I with symbols in S0(R, L−1;ℓ(cl) (X ; R;F, F ))

and S0(R, L−1;ℓ(cl) (X ; R;E,E)), respectively.


Moreover, the following are equivalent:• a is elliptic (on I).

• There exists b ∈ S−(R, L−µ;ℓ(cl) (X ; R;F,E)) such that ab − 1 and ba − 1

coincide in a neighbourhood of I with symbols in S−1(R, L−1;ℓ(cl) (X ; R;F, F ))

and S−1(R, L−1;ℓ(cl) (X ; R;E,E)), respectively.


with r0 ∈ R. Then the following are equivalent:• a is interior parabolic (on I).

• There exists b ∈ S−(R, L−µ;ℓV (cl)(X ; H;F,E)) such that ab − 1 and ba − 1

coincide in a neighbourhood of I with symbols in S0(R, L−1;ℓV (cl)(X ; H;F, F ))

and S0(R, L−1;ℓV (cl)(X ; H;E,E)), respectively.

Moreover, the following are equivalent:• a is parabolic (on I).

• There exists b ∈ S−(R, L−µ;ℓV (cl)(X ; H;F,E)) such that ab−1 and ba−1 co-

incide in a neighbourhood of I with symbols in S−1(R, L−1;ℓV (cl)(X ; H;F, F ))

and S−1(R, L−1;ℓV (cl)(X ; H;E,E)), respectively.

Proof. The first equivalences in a) and b) follow from Theorem 3.1.10 and Theorem3.2.18. It suffices to prove the necessity of the existence of b with the assertedproperties for the ellipticity or parabolicity on the interval I. We will concentrateon b) only, for the proof of a) is analogous.

Let a ∈ S(R, Lµ;ℓV (cl)(X ; H;E,F )) be parabolic on I. From the parabol-

icity in the interior we obtain together with Theorem 3.2.18 the existence of

b ∈ S−(R, L−µ;ℓV (cl)(X ; H;F,E)) and rL ∈ S0(R, L−∞

V (X ; H;E,E)) as well as

rR ∈ S0(R, L−∞V (X ; H;F, F )) such that ab = 1 + rR and ba = 1 + rL in a neigh-

bourhood of I. With a suitable excision function χ ∈ C∞(R) we define

bL := b− rLb+ rL(χ(r)a(r, ζ)−1

)rR,

bR := b− brR + rL(χ(r)a(r, ζ)−1

)rR.

Note that

rL(χ(r)a(r, ζ)−1

)rR ∈ S−(R, L−∞

V (X ; H;F,E))

by Definition 3.2.2 and the parabolicity of a: the function is analytic and rapidlydecreasing in H taking values in the bounded operators acting in the scale ofSobolev spaces on the manifold X ; the corresponding estimates in the variabler ∈ R are straightforward.

Consequently, we have that abR−1 and bLa−1 coincide with symbols belong-ing to S−∞(R, L−∞

V (X ; H;F, F )) and S−∞(R, L−∞V (X ; H;E,E)), respectively, in

a neighbourhood of I. Now we may choose either b = bL or b = bR to obtain thedesired properties.


Theorem 5.4.11. a) Let a ∈ S(R, Lµ;ℓ(cl)(X ; R;E,F )). Let either I = R or I =

[r0,∞), I = (−∞, r0] with r0 ∈ R. Then the following are equivalent:• a is interior elliptic on I.

• There exists a symbol b ∈ S−(R, L−µ;ℓ(cl) (X ; R;F,E)), and elements rR ∈

S0(R, L−∞(X ; R;F, F )), rL ∈ S0(R, L−∞(X ; R;E,E)), as well as func-tions χ, χ ∈ C∞(R) with χ, χ ≡ 1 on I and χ, χ ≡ 0 outside a neighbour-hood of I, such that

χ(opr(a)opr(b) − 1

)χ = opr(rR),

χ(opr(b)opr(a) − 1

)χ = opr(rL).

Moreover, the following are equivalent:• a is elliptic on I.

• There exists a symbol b ∈ S−(R, L−µ;ℓ(cl) (X ; R;F,E)), and elements rR ∈

S−∞(R, L−∞(X ; R;F, F )), rL ∈ S−∞(R, L−∞(X ; R;E,E)), as well asfunctions χ, χ ∈ C∞(R) with χ, χ ≡ 1 on I and χ, χ ≡ 0 outside a neigh-bourhood of I, such that


)χ = opr(rR),


)χ = opr(rL).


with r0 ∈ R. Then the following are equivalent:• a is interior parabolic on I.

• There exists a symbol b ∈ S−(R, L−µ;ℓV (cl)(X ; H;F,E)), and elements rR ∈

S0(R, L−∞V (X ; H;F, F )), rL ∈ S0(R, L−∞

V (X ; H;E,E)), as well as func-tions χ, χ ∈ C∞(R) with χ, χ ≡ 1 on I and χ, χ ≡ 0 outside a neighbour-hood of I, such that


)χ = opr(rR),


)χ = opr(rL).

Moreover, the following are equivalent:• a is parabolic on I.

• There exists a symbol b ∈ S−(R, L−µ;ℓV (cl)(X ; H;F,E)), and elements rR ∈

S−∞(R, L−∞V (X ; H;F, F )), rL ∈ S−∞(R, L−∞

V (X ; H;E,E)), as well asfunctions χ, χ ∈ C∞(R) with χ, χ ≡ 1 on I and χ, χ ≡ 0 outside a neigh-bourhood of I, such that


)χ = opr(rR),


)χ = opr(rL).

Proof. In view of Theorem 5.4.3, Theorem 5.4.4 and Lemma 5.4.10 the above con-ditions in a) and b) are clearly sufficient for the (interior) ellipticity or parabolicityof the symbols (on the interval I). For the proof of the necessity we restrict our-selves to consider b), and to the case I = R. Note first that the case of interior


parabolicity is analogous to Theorem 5.3.10, but now applied with similar argu-ments to the setting of Fourier operators with global weight conditions.

Now assume that the symbol a ∈ S(R, Lµ;ℓV (cl)(X ; H;E,F )) is parabolic.

From Lemma 5.4.10 and Theorem 5.4.4 we conclude that there exists b ∈S−(R, L−µ;ℓ

V (cl)(X ; H;F,E)) such that a#b = 1 − rR and b#a = 1 − rL with

rR ∈ S−1(R, L−1;ℓV (cl)(X ; H;F, F )),

rL ∈ S−1(R, L−1;ℓV (cl)(X ; H;E,E)).

Now choose rR, rL such that

S−1(R, L−1;ℓV (cl)(X ; H;F, F )) ∋ rR ∼

∞∑

j=1

#(j)rR,

S−1(R, L−1;ℓV (cl)(X ; H;E,E)) ∋ rL ∼

∞∑

j=1

#(j)rL.

These asymptotic expansions are to be understood in the following sense:

rR −N∑

j=1

#(j)rR ∈ S−1−N (R, L−1;ℓV (cl)(X ; H;F, F )),

rL −N∑

j=1

#(j)rL ∈ S−1−N (R, L−1;ℓV (cl)(X ; H;E,E)).

for N ∈ N0. Note that the existence of rR, rL with the corresponding asymp-totic expansions can be proved analogously to Section 2.1 by employing a Borel-argument with a 0-excision function in the variable r ∈ R involved. Now define

b := b#(1+rR) or b := (1+rL)#b. Then we see that a#b = 1−rR and b#a = 1−rLwith rR ∈ S−∞(R, L−1;ℓ

V (cl)(X ; H;F, F )) and rL ∈ S−∞(R, L−1;ℓV (cl)(X ; H;E,E)). Now

choose rR, rL such that

S−∞(R, L−1;ℓV (cl)(X ; H;F, F )) ∋ rR ∼

V

∞∑

j=1

#(j)rR,

S−∞(R, L−1;ℓV (cl)(X ; H;E,E)) ∋ rL ∼

V

∞∑

j=1

#(j)rL,

where the asymptotic expansions are to be carried out analogously to Theorem3.2.12 (with rapidly decreasing behaviour of the extra-parameter r ∈ R). Now we

see that if we define either b := b#(1 + rR) or b := (1 + rL)#b we obtain thedesired assertion, i. e.,

a#b− 1 ∈ S−∞(R, L−∞V (X ; H;F, F )),

b#a− 1 ∈ S−∞(R, L−∞V (X ; H;E,E)).

This finishes the proof of the theorem.


Chapter 6. Volterra cone calculus

6.1. Green operators

Remark 6.1.1. Throughout this chapter we again employ the notations from No-tation 3.1.1 with the corresponding data fixed on the manifold X and the vectorbundles E and F .

Definition 6.1.2. a) Let Θ = (θ, 0] with −∞ ≤ θ < 0, and let P ∈As

((γ,Θ), C∞(X,F )

)and Q ∈ As

((−γ,Θ), C∞(X,E)

)be asymptotic types.

Then an operator

G ∈ L(ind-lims,t,δ∈R

K(s,t),γ;ℓ(X∧, E)δ, proj-lims,t,δ∈R

K(s,t),γ;ℓ(X∧, E)δ)

is called a Green operator with respect to the asymptotic types P and Q, ifG and its formal adjoint G∗ with respect to the r−

n2 L2-inner product induce

continuous operators

G : ind-lims,t,δ∈R

K(s,t),γ;ℓ(X∧, E)δ −→ SγP (X∧, F ),

G∗ : ind-lims,t,δ∈R

K(s,t),−γ;ℓ(X∧, F )δ −→ S−γQ (X∧, E).

The space of all Green operators is denoted by CG(X∧, (γ,Θ);E,F ). If indica-tion of the concrete asymptotic types is necessary we emphasize this by writingCG(X∧, (γ,Θ);E,F )P,Q.

b) A Green operator G ∈ CG(X∧, (γ,Θ);E,F ) is called a Volterra Green operatorprovided that one of the following equivalent conditions is fulfilled:

• G restricts to continuous operators

G : H(s,t),γ;ℓ0 ((0, r0]×X,E) −→ H(s,t),γ;ℓ

0 ((0, r0]×X,F )

for every (some) s, t ∈ R and every r0 ∈ R+.• For every r0 ∈ R+ we have (Gu)(r) ≡ 0 for r > r0 for all u ∈C∞

0 (R+, C∞(X,E)) such that u(r) ≡ 0 for r > r0.

• For u ∈ L2,γ−n2 (R+, L

2(X,E)) and v ∈ L2,−γ−n2 (R+, L

2(X,F )) such thatsupp(u) < supp(v) we have 〈Gu, v〉

r−n2 L2 = 0.

The space of all Volterra Green operators is denoted by CG,V (X∧, (γ,Θ);E,F ),respectively CG,V (X∧, (γ,Θ);E,F )P,Q for the space of Volterra Green opera-tors with respect to the asymptotic types P and Q.

Remark 6.1.3. From Definition 6.1.2 we conclude that the class of (Volterra) Greenoperators is independent of the particular anisotropy ℓ ∈ N. Moreover, it forms anoperator algebra, i. e., if H is another vector bundle then the composition inducesa well-defined mapping

CG(,V )(X∧, (γ,Θ);F,H)×CG(,V )(X

∧, (γ,Θ);E,F ) −→ CG(,V )(X∧, (γ,Θ);E,H).

The class of Green operators is closed with respect to taking formal adjoints withrespect to the r−

n2 L2-inner product, i. e., the mapping

∗ : CG(X∧, (γ,Θ);E,F ) −→ CG(X∧, (−γ,Θ);F,E)


is well-defined.

Proposition 6.1.4. An operator G : C∞0 (R+, C

∞(X,E)) −→ D′(R+,D′(X,F ))belongs to CG(X∧, (γ,Θ);E,F )P,Q if and only if G can be represented both as

G(u) =

∞∑

j=1

λj〈u, xj〉r−n2 L2 sj ,

G(u) =

∞∑

j=1

λj〈u, sj〉r−n2 L2 xj ,

for u ∈ C∞0 (R+, C

∞(X,E)), where (λj), (λj) ∈ ℓ1, and (xj) ∈ K∞,−γ(X∧, E)∞,

(sj) ∈ SγP (X∧, F ), (xj) ∈ K∞,γ(X∧, F )∞, (sj) ∈ S−γQ (X∧, E) are sequences tend-

ing to zero in the corresponding spaces. Here we denote

K∞,γ(X∧, F )∞ =⋂

s,t,δ∈R

K(s,t),γ;ℓ(X∧, F )δ,

and analogously K∞,−γ(X∧, E)∞.In other words: G ∈ CG(X∧, (γ,Θ);E,F )P,Q if and only if

G ∈(K∞,−γ(X∧, E)∞⊗πSγP (X∧, F )

)∩

(S−γQ (X∧, E)⊗πK∞,γ(X∧, F )∞

).

In particular,

CG(X∧, (γ,Θ);E,F ) → S−γ(X∧, E)⊗πSγ(X∧, F )

→ ℓ1(K(s,t),γ;ℓ(X∧, E)δ,K(s′,t′),γ;ℓ(X∧, F )δ′)

for every s, s′, t, t′, δ, δ′ ∈ R.

Proof. By Theorem 4.3.4

K(s,t),γ;ℓ(X∧, E)n2+δ, r

−n2 L2(X∧, E),K(−s,−t),−γ;ℓ(X∧, E)n

2−δ and

K(s,t),γ;ℓ(X∧, F )n2 +δ, r

−n2 L2(X∧, F ),K(−s,−t),−γ;ℓ(X∧, F )n

2 −δare Hilbert triples for all s, t, δ ∈ R. Moreover, SγP (X∧, F ) and S−γ

Q (X∧, E) arenuclear Frechet spaces which are continuously embedded in the cone Sobolev spacesby Proposition 4.3.9. Let G ∈ CG(X∧, (γ,Θ);E,F )P,Q. From Proposition 1.3.9 weconclude that G belongs to⋂((

K(s,t),−γ;ℓ(X∧, E)δ⊗πSγP (X∧, F ))∩

(S−γQ (X∧, E)⊗πK(s′,t′),γ;ℓ(X∧, F )δ′

))

=(K∞,−γ(X∧, E)∞⊗πSγP (X∧, F )

)∩

(S−γQ (X∧, E)⊗πK∞,γ(X∧, F )∞

),

where the intersection is taken over all s, t, s′, t′, δ, δ′ ∈ R. The converse is imme-diate. We have

(K∞,−γ(X∧, E)∞⊗πSγP (X∧, F )

)∩

(S−γQ (X∧, E)⊗πK∞,γ(X∧, F )∞

)→

S−γ(X∧, E)⊗πSγ(X∧, F ) → ℓ1(K(s,t),γ;ℓ(X∧, E)δ,K(s′,t′),γ;ℓ(X∧, F )δ′)

for all s, s′, t, t′, δ, δ′ ∈ R. This finishes the proof of the proposition.


Proposition 6.1.5. Let G ∈ CG(X∧, (γ,Θ);E,F ) and ω, ω ∈ C∞0 (R+) be

cut-off functions near r = 0. Then we have (1 − ω)G(1 − ω) = opr(g∞)

with g∞ ∈ S−∞(R, L−∞(X ; R;E,F )), and ωGω = opγ−n

2

M (g0) with g0 ∈C∞B (R+, L

−∞(X ; Γn+12 −γ ;E,F )) such that limr→0 g0(r) = 0.

For G ∈ CG,V (X∧, (γ,Θ);E,F ) we have g∞ ∈ S−∞(R, L−∞V (X ; H;E,F ))

and g0 ∈ C∞B (R+, L

−∞V (X ; Hn+1

2 −γ ;E,F )).

Proof. From Proposition 6.1.4 we conclude that

(1 − ω)G(1 − ω) ∈ S(R×X,E)⊗πS(R×X,F ),

ωGω ∈ T−γ−n2(X∧, E)⊗πTγ−n

2(X∧, F ).

Consider the operator G∞ = (1 − ω)G(1 − ω): We may write

(G∞u

)(r) =

∫

R

k∞(r, r′)u(r′) dr′

for u ∈ C∞0 (R+, C

∞(X,E)) with a kernel k∞ ∈ S(R×R, L−∞(X ;E,F )). Let

χ ∈ S(R) such that∫R

χ(τ) dτ = 1, and set a(r, r′, τ) := e−i(r−r′)τk(r, r′)χ(τ).

Then a ∈ S−∞,−∞(R × R, L−∞(X ; R;E,F )) is a double-symbol in the Fouriercalculus with global weight conditions from Section 5.4, and we haveG∞ = opr(a).Thus we obtain g∞ as the left-symbol aL according to Theorem 5.4.3, i. e.,

g∞(r, τ) =

∫∫e−ir

′ηa(r, r + r′, τ + η) dr′ dη =

∫

R

e−ir′τk∞(r, r − r′) dr′.

If G ∈ CG,V (X∧, (γ,Θ);E,F ) we have k∞(r, r − r′) ≡ 0 for r′ > 0, and con-sequently g∞ extends as an analytic rapidly decreasing function to the upperhalf-plane in view of the Paley–Wiener theorem (see also Section 1.1), i. e.,g∞ ∈ S−∞(R, L−∞

V (X ; H;E,F )) as desired.Now consider the operator G0 = ωGω: We may write

(G0u

)(r) =

∫

R+

k0(r, r′)u(r′)

dr′

r′

for u ∈ C∞0 (R+, C

∞(X,E)) with a kernel

k0 ∈ Tγ−n2(R+)⊗πT−γ+n

2+1(R+)⊗πL−∞(X ;E,F ).

Let χ ∈ S(Γn+12 −γ) such that 1

2πi

∫Γn+1

2−γ

χ(z) dz = 1. Then G0 = opγ−n

2

M (b) with

the Mellin double-symbol b(r, r′, z) :=(rr′

)zk0(r, r

′)χ(z). From Theorem 5.3.2 we


obtain G0 = opγ−n

2

M (g0) with

g0(r, z) =

∫

R

∫

R+

siηb(r, sr, z + iη)ds

sdη

=

∫

R+

szk0(r, rs−1)

ds

s.

We have limr→0 g0(r) = 0 since 〈log(r)〉g0(r) is bounded as r → 0.If G ∈ CG,V (X∧, (γ,Θ);E,F ) we conclude that k0(r, rs

−1) ≡ 0 for s > 1, andconsequently g0 ∈ C∞

B (R+, L−∞V (X ; Hn+1

2 −γ ;E,F )) in view of the Paley–Wiener

theorem (see Section 1.1).

Theorem 6.1.6. a) Let G ∈ CG(X∧, (γ,Θ);E) such that 1 + G is invertiblein L(K(s,t),γ;ℓ(X∧, E)δ) for some s, t, δ ∈ R. Then 1 + G is invertible in

L(K(s,t),γ;ℓ(X∧, E)δ) for all s, t, δ ∈ R, and the inverse is given as(1+G

)−1=

1 +G1 with a Green operator G1 ∈ CG(X∧, (γ,Θ);E).b) Let G ∈ CG,V (X∧, (γ,Θ);E). Then 1 +G is invertible in L(K(s,t),γ;ℓ(X∧, E)δ)

for all s, t, δ ∈ R, and we have(1 + G

)−1= 1 + G1 with a Volterra Green

operator G1 ∈ CG,V (X∧, (γ,Θ);E).

Proof. For the proof of a) note that we may write(1 +G

)−1= 1 −G+G

(1 +G

)−1G.

The operator G1 := −G+G(1 +G

)−1G fulfills the conditions in Definition 6.1.2,

and consequently belongs to CG(X∧, (γ,Θ);E). Clearly, 1 + G1 inverts 1 + G inL(K(s,t),γ;ℓ(X∧, E)δ) for all s, t, δ ∈ R.

Let us now prove b). We first consider the weight γ = n2 . Since we have

(1 −G)(1 +G) = (1 +G)(1 −G) = 1 −G2,

where G2 ∈ CG,V (X∧, (γ,Θ);E), we just have to prove the assertion for the oper-ator 1 −G2. We may write G2 = G(G∗)∗. Thus Proposition 1.3.9 gives

G2 ∈ S−n2

Q (X∧, E)⊗πSn2

P (X∧, E)

with suitable asymptotic types P and Q, i. e., G2 has a representation

G2(u) =∞∑

j=1

λj〈u, sj〉r−n2 L2 sj

for u ∈ C∞0 (R+, C

∞(X,E)) with (λj) ∈ ℓ1 and sequences (sj) ∈ S−n2

Q (X∧, E) and

(sj) ∈ Sn2

P (X∧, E) tending to zero. In particular, we have

(G2u

)(r) =

∫

R+

k(r, r′)u(r′) dr′


for u ∈ L2(R+, L2(X,E)) = K(0,0),n2 ;ℓ(X∧, E)0 with a Volterra integral kernel

k ∈ C(R+×R+,L(L2(X,E))) that satisfies

supg(r)g(r′)‖k(r, r′)‖L(L2(X,E)); r, r′ ∈ R+ <∞.

Here g ∈ C(R+) is a function of the form

g(r) = ω(r)rε + (1 − ω(r))r

with a cut-off function ω ∈ C∞0 (R+) near r = 0 and a sufficiently small 0 < ε < 1

2 ,and thus we have ∫

R+

1

g(r)2dr <∞.

Consequently, we may apply Theorem 1.3.6 to the Volterra integral operator G2 ∈L(L2(R+, L

2(X,E))) and conclude that G2 is quasinilpotent. Moreover, we may

write(1 − G2

)−1= 1 − G1 with a Volterra integral operator G1. By a) G1 ∈

CG,V (X∧, (n2 ,Θ);E) is a Volterra Green operator, and we have(1 −G2

)−1= 1 −G1 ∈ L(K(s,t),n2 ;ℓ(X∧, E)δ)

for all s, t, δ ∈ R. This finishes the proof for the weight γ = n2 .

Next consider the case of general weights γ ∈ R. We may write

1 +G = rγ−n2

(1 + G

)r−(γ−n

2 ) ∈ L(K(s,t),γ;ℓ(X∧, E)δ)

for s, t, δ ∈ R, where G := r−(γ−n2 )Grγ−

n2 is a Volterra Green operator in

CG,V (X∧, (n2 ,Θ);E). From the first part of the proof we conclude that 1 + G

is invertible, and we have(1 + G

)−1= 1 + G1 with a Volterra Green op-

erator G1 ∈ CG,V (X∧, (n2 ,Θ);E). Thus also 1 + G is invertible with inverse(1 + G

)−1= 1 + G1, where G1 := rγ−

n2 G1r

−(γ−n2 ) is a Volterra Green opera-

tor in the space CG,V (X∧, (γ,Θ);E). This completes the proof of the theorem.

Corollary 6.1.7. Let G ∈ CG,V (X∧, (γ,Θ);E). Then 1 + G restricts to an iso-morphism

1 +G : H(s,t),γ;ℓ0 ((0, r0]×X,E) −→ H(s,t),γ;ℓ

0 ((0, r0]×X,E)

for all s, t ∈ R and every r0 ∈ R+, and we have(1 + G

)−1= 1 + G1 with

G1 ∈ CG,V (X∧, (γ,Θ);E).

6.2. The algebra of conormal operators

Operators that generate asymptotics.

Definition 6.2.1. Let γ1, γ2 ∈ R, and let Θ = (θ, 0] with −∞ ≤ θ < 0. We definespaces of (Volterra) operators that generate asymptotics as follows:

a) An operator G belongs to Cµ,;ℓG (X∧, (γ1, γ2,Θ);E,F ) for µ, ∈ R, if the fol-lowing conditions are fulfilled:


• G and the formal adjoint G∗ with respect to the r−n2 L2-inner product are

well-defined as continuous operators

G : K(s,t),γ1;ℓ(X∧, E)δ −→ K(s−µ,t),γ2;ℓ(X∧, F )δ−,

G∗ : K(s,t),−γ2;ℓ(X∧, F )δ −→ K(s−µ,t),−γ1;ℓ(X∧, F )δ−

for all s, t, δ ∈ R.• There exist asymptotic types P ∈ As

((γ2,Θ), C∞(X,F )

)and Q ∈

As((−γ1,Θ), C∞(X,E)

)such that

G :

K(s,t),γ1;ℓ(X∧, E)δ −→ K(s−µ,t),γ2;ℓ

P (X∧, F )δ−,

Sγ1(X∧, E) −→ Sγ2P (X∧, F ),

G∗ :

K(s,t),−γ2;ℓ(X∧, F )δ −→ K(s−µ,t),−γ1;ℓ

Q (X∧, E)δ−,

S−γ2(X∧, F ) −→ S−γ1Q (X∧, E),

for all s, t, δ ∈ R.

b) Let Cµ,;ℓG,V (X∧, (γ1, γ2,Θ);E,F ) denote the subspace of all operators G ∈Cµ,;ℓG (X∧, (γ1, γ2,Θ);E,F ) such that one of the following equivalent condi-tions is fulfilled:

• G restricts to continuous operators

G : H(s,t),γ1;ℓ0 ((0, r0]×X,E) −→ H(s−µ,t),γ2;ℓ

0 ((0, r0]×X,F )

for every (some) s, t ∈ R and every r0 ∈ R+.• For every r0 ∈ R+ we have (Gu)(r) ≡ 0 for r > r0 for all u ∈C∞

0 (R+, C∞(X,E)) such that u(r) ≡ 0 for r > r0.

Remark 6.2.2. a) Definition 6.2.1 implies that the spaces of (Volterra) operatorsthat generate asymptotics form (bi-)graded operator algebras, i. e., if H isanother vector bundle then the composition induces a well-defined mapping

Cµ′,′;ℓ

G(,V ) (X∧, (γ2, γ3,Θ);F,H)×Cµ,;ℓG(,V )(X∧, (γ1, γ2,Θ);E,F )

−→ Cµ+µ′,+′;ℓG(,V ) (X∧, (γ1, γ3,Θ);E,H).

Moreover, taking formal adjoints with respect to the r−n2 L2-inner product in-

duces a well-defined mapping

∗ : Cµ,;ℓG (X∧, (γ1, γ2,Θ);E,F ) −→ Cµ,;ℓG (X∧, (−γ2,−γ1,Θ);F,E),

i. e., the operators that generate asymptotics form a (bi-)graded ∗-algebra.


b) For µ, ∈ R we denote

Cµ;ℓG(,V )(X

∧, (γ1, γ2,Θ);E,F ) :=⋂

′∈R

Cµ,′ ;ℓ

G(,V )(X∧, (γ1, γ2,Θ);E,F ),

CG(,V )(X∧, (γ1, γ2,Θ);E,F ) :=

⋂

µ′∈R

Cµ′,;ℓ

G(,V )(X∧, (γ1, γ2,Θ);E,F ),

CG(,V )(X∧, (γ1, γ2,Θ);E,F ) :=

⋂

µ′,′∈R

Cµ′,′;ℓ

G(,V ) (X∧, (γ1, γ2,Θ);E,F ).

The elements of the latter of these spaces are also called (Volterra) Greenoperators associated with the (double) weight datum (γ1, γ2,Θ). Indeed, if γ1 =γ2 we have

CG(,V )(X∧, (γ1, γ2,Θ);E,F ) = CG(,V )(X

∧, (γ1,Θ);E,F )

according to Definition 6.1.2.c) The (Volterra) Green operators form a two-sided ideal in the algebra of

(Volterra) operators that generate asymptotics.d) If γ1 = γ2 we simplify the notations by substituting (γ1,Θ) for (γ1, γ2,Θ).

Calculus of conormal symbols.

Definition 6.2.3. a) Let (γ, (−N, 0]) be a weight datum, N ∈ N. For µ ∈ R

define the space of (classical) conormal symbols with respect to the weightdatum (γ, (−N, 0]) as

Σµ;ℓM(cl)(X, (γ,(−N, 0]);E,F ) := (h0, . . . , hN−1);

hj ∈Mµ;ℓPj(cl)

(X ;E,F ), πCP0∩Γn+12 −γ = ∅.

(6.2.1)

The subspace of (classical) Volterra conormal symbols with respect to theweight datum (γ, (−N, 0]) is defined as

Σµ;ℓM,V (cl)(X, (γ,(−N, 0]);E,F ) := (h0, . . . , hN−1);

hj ∈Mµ;ℓV,Pj(cl)

(X ; Hn+12 −γ+j;E,F ).

(6.2.2)

We define the spaces of order −∞ as

ΣM (X, (γ, (−N, 0]);E,F ) :=⋂

µ∈R

Σµ;ℓM (X, (γ, (−N, 0]);E,F ),

ΣM,V (X, (γ, (−N, 0]);E,F ) :=⋂

µ∈R

Σµ;ℓM,V (X, (γ, (−N, 0]);E,F ).

These spaces do not depend on the anisotropy ℓ ∈ N, and they consist of allN -tuples of meromorphic (Volterra) Mellin symbols of order −∞ with the sameconditions on the Mellin asymptotic types as above.


b) Let G be another vector bundle over X . We define the Mellin translation prod-uct

# : Σµ;ℓM (X, (γ, (−N, 0]);F,G)×Σµ

′;ℓM (X, (γ, (−N, 0]);E,F )

−→ Σµ+µ′;ℓM (X, (γ, (−N, 0]);E,G),

(g0, . . . , gN−1)#(h0, . . . , hN−1) := (h0, . . . , hN−1),

hk :=∑

p+q=k

(T−qgp)(hq),

(6.2.3)

where T denotes the translation operator for functions in the complex plane,i. e.,

((T−qgp)(hq)

)(z) = gp(z − q)hq(z).

c) We define a ∗-operation via

∗ : Σµ;ℓM (X, (γ, (−N, 0]);E,F ) −→ Σµ;ℓ

M (X, (−γ, (−N, 0]);F,E),

(h0, . . . , hN−1)∗ := (h0, . . . , hN−1),

hk(z) :=(hk(n+ 1 − k − z)

)(∗),

(6.2.4)

where (∗) denotes the formal adjoint with respect to the L2-inner product onthe manifold.

Theorem 6.2.4. a) The spaces of (classical) conormal symbols form a graded ∗-algebra with componentwise linear operations, the Mellin translation product(6.2.3), and the ∗-operation (6.2.4). The conormal symbols of order −∞ forma symmetric two-sided ideal.

More precisely, this means the following: Let E,F,G,H ∈ Vect(X) be com-plex vector bundles with corresponding data fixed according to Notation 3.1.1.

i) Σµ;ℓM(cl)(X, (γ, (−N, 0]);E,F ) is a vector space with componentwise addition

and scalar multiplication.ii) The Mellin translation product induces an associative product, i. e., it is

well-defined as a bilinear mapping

# : Σµ;ℓM(cl)(X, (γ, (−N, 0]);F,G)×Σµ

′;ℓM(cl)(X, (γ, (−N, 0]);E,F )

−→ Σµ+µ′;ℓM(cl) (X, (γ, (−N, 0]);E,G),

and we have (a#b)#c = a#(b#c) ∈ Σµ1+µ2+µ3;ℓM(cl) (X, (γ, (−N, 0]);E,H) for

a ∈ Σµ3;ℓM(cl)(X, (γ, (−N, 0]);G,H), b ∈ Σµ2;ℓ

M(cl)(X, (γ, (−N, 0]);F,G), and

c ∈ Σµ1;ℓM(cl)(X, (γ, (−N, 0]);E,F ).

iii) The ∗-operation is well-defined as an antilinear mapping

∗ : Σµ;ℓM (X, (γ, (−N, 0]);E,F ) −→ Σµ;ℓ

M (X, (−γ, (−N, 0]);F,E),

and we have (a#b)∗ = b∗#a∗, (a∗)∗ = a for conormal symbols a ∈Σµ;ℓM(cl)(X, (γ, (−N, 0]);F,G) and b ∈ Σµ

′;ℓM(cl)(X, (γ, (−N, 0]);E,F ).


b) The spaces of (classical) Volterra conormal symbols form a graded subalgebra,i. e., they share the properties i) and ii) listed in a) with ΣM(cl) replaced byΣM,V (cl). Note that they are not closed with respect to the ∗-operation.

The Volterra conormal symbols of order −∞ form a two-sided ideal.

Proof. These assertions follow via simple algebraic calculations from Theorem5.1.8 and Theorem 5.2.4.

Definition 6.2.5. a) a = (h0, . . . , hN−1) ∈ Σµ;ℓM(cl)(X, (γ, (−N, 0]);E,F ) is

called elliptic if

• h0 is elliptic as an element of Mµ;ℓP0(cl)

(X ;E,F ) in the sense of Definition

5.1.12,• there exists s0 ∈ R such that h0(z) : Hs0(X,E) −→ Hs0−µ(X,F ) is

bijective for all z ∈ Γn+12 −γ .

b) An element a = (h0, . . . , hN−1) ∈ Σµ;ℓM,V (cl)(X, (γ, (−N, 0]);E,F ) is called par-

abolic if• h0 is parabolic as an element of Mµ;ℓ

V,P0(cl)(X ; Hn+1

2 −γ ;E,F ) in the sense

of Definition 5.2.7,• there exists s0 ∈ R such that h0(z) : Hs0(X,E) −→ Hs0−µ(X,F ) is

bijective for all z ∈ Hn+12 −γ .

Notation 6.2.6. Let E be any vector bundle over X . For the moment, we prefer todenote the unit with respect to the Mellin translation product as

1 := (1, 0, . . . , 0) ∈ Σ0;ℓM,V (cl)(X, (γ, (−N, 0]);E).

Theorem 6.2.7. a) Let a ∈ Σµ;ℓM(cl)(X, (γ, (−N, 0]);E,F ). Then the following are

equivalent:i) a is elliptic in the sense of Definition 6.2.5.ii) a is invertible within the algebra of conormal symbols, i. e., there exists

b ∈ Σ−µ;ℓM(cl)(X, (γ, (−N, 0]);F,E) such that

a#b = 1 ∈ Σ0;ℓM(cl)(X, (γ, (−N, 0]);F ),

b#a = 1 ∈ Σ0;ℓM(cl)(X, (γ, (−N, 0]);E).

b) Let a ∈ Σµ;ℓM,V (cl)(X, (γ, (−N, 0]);E,F ). Then the following are equivalent:

i) a is parabolic in the sense of Definition 6.2.5.ii) a is invertible within the algebra of Volterra conormal symbols, i. e., there

exists b ∈ Σ−µ;ℓM,V (cl)(X, (γ, (−N, 0]);F,E) such that

a#b = 1 ∈ Σ0;ℓM,V (cl)(X, (γ, (−N, 0]);F ),

b#a = 1 ∈ Σ0;ℓM,V (cl)(X, (γ, (−N, 0]);E).


Proof. By Theorem 5.1.14 and Theorem 5.2.8 the conditions ii) in a) and b) aresufficient for the ellipticity (parabolicity) of a in view of the definition of the Mellintranslation product.

It remains to show the necessity. Let a = (h0, . . . , hN−1). We define thecomponents of b = (g0, . . . , gN−1) by induction as follows:

By Theorem 5.1.14 and Theorem 5.2.8 a is elliptic, respectively parabolic,

if and only if there exists g0 ∈ M−µ;ℓQ0(cl)

(X ;F,E), πCQ0∩Γn+12 −γ = ∅, respectively

g0 ∈ M−µ;ℓV,Q0(cl)

(X ; Hn+12 −γ ;F,E), such that h0g0 ≡ 1 and g0h0 ≡ 1. Assume we

have already constructed g0, . . . , gk−1 for some k < N . Define

gk := −(T−kg0)∑

p+q=kq<k

(T−qhp)gq ∈M−µ;ℓQk(cl)

(X ;F,E)

M−µ;ℓV,Qk(cl)

(X ; Hn+12 −γ+k;F,E),

which is well-defined in view of Theorem 5.1.8 and Theorem 5.2.4. By constructionwe at once have a#b = 1, and a short calculation reveals b#a = 1. This finishesthe proof of the theorem.

Corollary 6.2.8. Let a ∈ ΣM(,V )(X, (γ, (−N, 0]);E), where a = (h0, . . . , hN−1).Then the following are equivalent:

a) 1 + a is elliptic (parabolic).b) There exists s0 ∈ R such that 1 + h0(z) ∈ L(Hs0(X,E)) is bijective for all

z ∈ Γn+12 −γ, respectively z ∈ Hn+1

2 −γ.

c) 1 + a is invertible within Σ0;ℓM(,V )(X, (γ, (−N, 0]);E) with respect to the Mellin

translation product, and the inverse is given as (1 + a)−1 = 1 + b with b ∈ΣM(,V )(X, (γ, (−N, 0]);E).

Proof. Due to Theorem 6.2.7 we just have to prove that the inverse of 1 + a in c)is of the asserted form. But this follows from the identity

(1 + a)−1 = 1 − a+ a#(1 + a)−1a.

Note that b := −a + a#(1 + a)−1a ∈ ΣM(,V )(X, (γ, (−N, 0]);E) since we handlewith a two-sided ideal.

The operator calculus.

Definition 6.2.9. For µ, ∈ R the set Cµ,;ℓM+G(cl)(X∧, (γ, (−N, 0]);E,F ) of (clas-

sical) conormal operators with respect to the weight datum (γ, (−N, 0]), N ∈ N,consists of all operators A : Sγ(X∧, E) −→ Sγ(X∧, F ) of the form

A =N−1∑

j=0

ωjrjop

γjM (hj)ωj +G (6.2.5)

with an operator G ∈ Cµ,;ℓG (X∧, (γ, (−N, 0]);E,F ) and

i) cut-off functions ωj , ωj ∈ C∞0 (R+) near r = 0,

ii) γ − n2 − j ≤ γj ≤ γ − n

2 ,


iii) meromorphic Mellin symbols hj ∈Mµ;ℓPj(cl)

(X ;E,F ) such that πCPj∩Γ 12−γj

=

∅.The subset Cµ,;ℓM+G,V (cl)(X

∧, (γ, (−N, 0]);E,F ) of (classical) Volterra conormal op-

erators with respect to the weight datum (γ, (−N, 0]) consists of all those operators

A having a representation as in (6.2.5) with G ∈ Cµ,;ℓG,V (X∧, (γ, (−N, 0]);E,F ), and

hj ∈Mµ;ℓV,Pj(cl)

(X ; H 12−γj

;E,F ).

Note that the meromorphic Mellin symbols (h0, . . . , hN−1) of a conormal

operator A ∈ Cµ,;ℓM+G(,V (cl))(X∧, (γ, (−N, 0]);E,F ) give rise to an element in

Σµ;ℓM(,V (cl))(X, (γ, (−N, 0]);E,F ).

Remark 6.2.10. The elements of Cµ,;ℓM+G(cl)(X∧, (γ, (−N, 0]);E,F ) are indeed well-

defined as continuous operators Sγ(X∧, E) −→ Sγ(X∧, F ). In fact, every sum-mand is continuous in

ωjrjop

γjM (hj)ωj : Sγj+n

2 (X∧, E) −→ Sγj+n2 +j(X∧, F ),

and we have Sγ(X∧, E) → Sγj+n2 (X∧, E) and Sγj+n

2 +j(X∧, F ) → Sγ(X∧, F ).

Moreover, an operator A ∈ Cµ,;ℓM+G,V (cl)(X∧, (γ, (−N, 0]);E,F ) restricts to a

continuous operator

A : Tγ−n2 ,0

((0, r0), C∞(X,E)) −→ Tγ−n

2 ,0((0, r0), C

∞(X,F ))

for every r0 ∈ R+. This follows from Proposition 5.3.5 and Definition 6.2.1.

Proposition 6.2.11. Let h ∈ Mµ;ℓR (X ;E,F ), and let πCR∩Γn+1

2 −γ = ∅. Then

opγ−n

2

M (h) extends by continuity to an operator

opγ−n

2

M (h) : H(s,t),γ;ℓ(X∧, E) −→ H(s−µ,t),γ;ℓ(X∧, F )

for every s, t ∈ R.Let Θ = (−θ, 0], where −∞ ≤ θ < 0. For every asymptotic type P ∈

As((γ,Θ), C∞(X,E)) there exists an asymptotic type Q ∈ As((γ,Θ), C∞(X,F ))

such that opγ−n

2

M (h) restricts to continuous operators

opγ−n

2

M (h) :

H(s,t),γ;ℓP (X∧, E) −→ H(s−µ,t),γ;ℓ

Q (X∧, F ),

Tγ−n2 ,P

(X∧, E) −→ Tγ−n2 ,Q

(X∧, F ).

Proof. The first assertion follows from Theorem 5.3.6. Due to Corollary 5.1.11 we

have Mµ;ℓR (X ;E,F ) = Mµ;ℓ

O (X ;E,F ) +M−∞R (X ;E,F ). Consequently, the proof

of the second assertion reduces to consider the cases h ∈ Mµ;ℓO (X ;E,F ) and h ∈

M−∞R (X ;E,F ).

The characterization of the Mellin image of Tγ−n2 ,P

(X∧, E) from Theorem4.2.16 shows that h acts as a multiplier in the spaces

h : Mγ−n2

(Tγ−n

2,P (X∧, E)

)−→ Mγ−n

2

(Tγ−n

2,Q(X∧, F )

)


with a certain asymptotic type Q both in the cases h ∈ Mµ;ℓO (X ;E,F ) and h ∈

M−∞R (X ;E,F ), i. e.,

opγ−n

2

M (h) : Tγ−n2 ,P

(X∧, E) −→ Tγ−n2 ,Q

(X∧, F )

as asserted.We have H(s,t),γ;ℓ

P (X∧, E) = H(s,t),γ;ℓΘ (X∧, E) + Tγ−n

2,P (X∧, E) by Theorem

4.2.16, and thus the remaining proof reduces to consider the case of the empty

asymptotic type, i. e., P = Θ. Let h ∈Mµ;ℓO (X ;E,F ), and let

Rs(τ) ∈ Ls;ℓ(X ; R;E),

Rs−µ(τ) ∈ Ls−µ;ℓ(X ; R;F )

be parameter-dependent reductions of orders from Theorem 3.1.12. Then we have

Rs−µ(τ)h(β + iτ) =(Rs−µ(τ)h(β + iτ)R−s(τ)

)

︸︷︷︸∈ C∞(Rβ , S

0(Rτ ;Ht(X,E), Ht(X,F )))

Rs(τ),

and thus h acts as a multiplier in the spaces

h : Mγ−n2

(H(s,t),γ;ℓ

Θ (X∧, E))−→ Mγ−n

2

(H(s−µ,t),γ;ℓ

Θ (X∧, F ))

due to Theorem 4.2.16.Now let h ∈M−∞

R (X ;E,F ), and let χ ∈ C∞(C) be an arbitrary πCR-excisionfunction. Then χ(β + iτ)h(β + iτ) ∈ C∞(Rβ ,S(Rτ , L

−∞(X ;E,F ))). Hence The-orem 4.2.16 implies that h acts as a multiplier in the spaces

h : Mγ−n2

(H(s,t),γ;ℓ

Θ (X∧, E))−→ Mγ−n

2

(H∞,γQ (X∧, F )

)

with a certain asymptotic type Q such that πCQ = πCR. Summing up, we haveshown that

opγ−n

2

M (h) : H(s,t),γ;ℓΘ (X∧, E) −→ H(s−µ,t),γ;ℓ

Q (X∧, F )

both in the cases h ∈ Mµ;ℓO (X ;E,F ) and h ∈ M−∞

R (X ;E,F ). This finishes theproof of the proposition.

Theorem 6.2.12. Let A ∈ Cµ,;ℓM+G(cl)(X∧, (γ, (−N, 0]);E,F ).

a) A extends by continuity to an operator


for all s, t, δ ∈ R.Moreover, for every asymptotic type P ∈ As((γ, (−N, 0]), C∞(X,E)) there ex-ists an asymptotic type Q ∈ As((γ, (−N, 0]), C∞(X,F )) such that A restrictsto continuous operators

A :

K(s,t),γ;ℓP (X∧, E)δ −→ K(s−µ,t),γ;ℓ

Q (X∧, F )δ−

SγP (X∧, E) −→ SγQ(X∧, F )



Let A ∈ Cµ,;ℓM+G,V (X∧, (γ, (−N, 0]);E,F ). Then A restricts for every r0 ∈ R+

to continuous operators

A : H(s,t),γ;ℓ0 ((0, r0]×X,E) −→ H(s−µ,t),γ;ℓ

0 ((0, r0]×X,F )

for all s, t ∈ R.b) The formal adjoint A∗ with respect to the r−

n2 L2-inner product belongs to

Cµ,;ℓM+G(cl)(X∧, (−γ, (−N, 0]);F,E). More precisely, let

A =

N−1∑

j=0

ωjrjop

γjM (hj)ωj +G

be a representation of A from (6.2.5). Then

A∗ =

N−1∑

j=0

ωjrjop

−γj−n−jM (hj)ωj +G∗,

hj(z) := hj(n+ 1 − j − z)(∗) ∈Mµ;ℓQj(cl)

(X ;F,E),

is a representation of A∗ in the sense of (6.2.5), where (∗) denotes the formaladjoint with respect to the L2-inner product on the manifold.

Proof. From Theorem 5.3.6 we obtain that every summand

ωjrjop

γjM (hj)ωj : K(s,t),γj+

n2 ;ℓ(X∧, E)δ −→ K(s−µ,t),γj+

n2 +j;ℓ(X∧, F )∞

in the representation of A from (6.2.5) is continuous for all s, t, δ ∈ R, and we have

K(s,t),γ;ℓ(X∧, E)δ → K(s,t),γj+n2 ;ℓ(X∧, E)δ,

K(s−µ,t),γj+n2 +j;ℓ(X∧, F )∞ → K(s−µ,t),γ;ℓ(X∧, F )∞.

This proves the first assertion in a). The continuity of A in the subspaces withasymptotics follows from Proposition 6.2.11.

If A ∈ Cµ,;ℓM+G,V (X∧, (γ, (−N, 0]);E,F ), then


0 ((0, r0]×X,F )

is continuous due to Proposition 5.3.5 and Theorem 5.3.6, respectively.Let us now prove b). Using Theorem 5.3.3 and Proposition 2.6.4 we may

write(ωjr

jopγjM (hj)ωj

)∗= ωjop

−γj−nM (hj(n+ 1 − z)(∗))rjωj

= ωjrjop

−γj−n−jM (hj)ωj .


Lemma 6.2.13. Let h ∈ Mµ;ℓP (X ;E,F ), and let ω, ω ∈ C∞

0 (R+) be cut-offfunctions. Moreover, let γ1, γ2 ∈ R, γ1 < γ2, such that πCP∩Γ 1

2−γ1= ∅ and

πCP∩Γ 12−γ2

= ∅.


Then the operator

ωopγ1M (h)ω − ωopγ2M (h)ω ∈ CG(X∧, (γ2 +n

2, γ1 +

n

2, (−∞, 0]);E,F ),

and it is finite-dimensional. If πCP∩Γ( 12−γ2,

12−γ1) = ∅ then the operator is identi-

cally zero.

Proof. Using the residue theorem we may write(opγ1M (h)u

)(r) −

(opγ2M (h)u

)(r) =

∑

p∈πCP∩ΓI

resp(r−zh(z)(Mu)(z)

)

for u ∈ C∞0 (R+, C

∞(X,E)), where I := (12 − γ2,

12 − γ1). Let (p,m,L) ∈ P such

that p ∈ πCP∩ΓI , and set

Up := m∑

k=0

cp,kr−p logk(r); cp,k ∈ 〈L(C∞(X,E))〉 ⊆ C∞(X,F )

.

Then we have resp(r−zh(z)(Mu)(z)

)∈ Up, i. e.,

(ωopγ1M (h)ω − ωopγ2M (h)ω

)(C∞

0 (R+, C∞(X,E))

)⊆ ω

∑

p∈πCP∩ΓI

Up = EQ(X∧, F )

with the induced asymptotic type Q ∈ As((γ1 + n

2 , (−∞, 0]), C∞(X,F )). Since

C∞0 (R+, C

∞(X,E)) is dense in K(s,t),γ2+n2 ;ℓ(X∧, E)δ for all s, t, δ ∈ R, and

EQ(X∧, F ) is finite-dimensional, we conclude

ωopγ1M (h)ω − ωopγ2M (h)ω : K(s,t),γ2+n2 ;ℓ(X∧, E)δ −→ EQ(X∧, F ).

Theorem 5.3.3 implies(ωopγ1M (h)ω − ωopγ2M (h)ω

)∗= ωop−γ1−n

M (h)ω − ωop−γ2−nM (h)ω

with h(z) := h(n+ 1 − z)(∗), where (∗) denotes the formal adjoint with respect tothe L2-inner product on the manifold, and thus we obtain with the same reasoningas above

(ωopγ1M (h)ω − ωopγ2M (h)ω

)∗: K(s,t),−γ1−

n2 ;ℓ(X∧, F )δ −→ EQ(X∧, E)

for all s, t, δ ∈ R with an asymptotic type Q ∈ As((−γ2− n

2 , (−∞, 0]), C∞(X,E)).

Remark 6.2.14. In the notation from Lemma 6.2.13 assume furthermore that γ −n2 − j ≤ γ1, γ2 ≤ γ − n

2 for some γ ∈ R and j ∈ N0. Then we conclude that

ωrjopγ1M (h)ω − ωrjopγ2M (h)ω ∈ CG(X∧, (γ, (−∞, 0]);E,F ),

and it is finite-dimensional. If πCP∩Γ( 12−γ2,

12−γ1)

= ∅, then the operator is identi-

cally zero.


Lemma 6.2.15. Let ω, ω ∈ C∞0 (R+) be cut-off functions, and let ϕ, ψ ∈ C∞

0 (R+).

Moreover, let γ − n2 − j ≤ γj ≤ γ − n

2 , j ∈ N0, and h ∈ Mµ;ℓP (X ;E,F ) such

that πCP∩Γ 12−γj

= ∅, respectively h ∈ Mµ;ℓV,P (X ; H 1

2−γj;E,F ). Then the following

holds:

a) ωrjopγjM (h)ϕ, ψrjop

γjM (h)ω, ψrjop

γjM (h)ϕ ∈ Cµ;ℓ

G(,V )(X∧, (γ, (−∞, 0]);E,F ).

b) If j > 0 then ωrjopγjM (h)ω ∈ Cµ;ℓ

G(,V )(X∧, (γ, (−j, 0]);E,F ).

Proof. Let ω ∈ C∞0 (R+) be a cut-off function such that ωϕ ≡ ϕ, and write

ωrjopγjM (h)ϕ =

(ωrjop

γjM (h)ω

)ϕ.

For every s, t, δ ∈ R we have

ϕ :

Sγ(X∧, E) −→ Sγj+

n2

(−∞,0](X∧, E),

K(s,t),γ;ℓ(X∧, E)δ −→ K(s,t),γj+n2 ;ℓ

(−∞,0] (X∧, E)∞,

ωrjopγjM (h)ω :

Sγj+

n2

(−∞,0](X∧, E) −→ SγQ(X∧, F ),

K(s,t),γj+n2 ;ℓ

(−∞,0] (X∧, E)∞ −→ K(s−µ,t),γ;ℓQ (X∧, F )∞

with a certain asymptotic type Q ∈ As((γ, (−∞, 0]), C∞(X,F )

)due to Proposi-

tion 6.2.11, i. e.,

ωrjopγjM (h)ϕ :

Sγ(X∧, E) −→ SγQ(X∧, F ),

K(s,t),γ;ℓ(X∧, E)δ −→ K(s−µ,t),γ;ℓQ (X∧, F )∞.

Moreover, we have

ψrjopγjM (h)ω, ψrjop

γjM (h)ϕ :

Sγ(X∧, E) −→ Sγ(−∞,0](X

∧, F ),

K(s,t),γ;ℓ(X∧, E)δ −→ K(s−µ,t),γ;ℓ(−∞,0] (X∧, F )∞.

Theorem 6.2.12 implies(ωrjop

γjM (h)ϕ

)∗= ϕrjop

−γj−n−jM (h)ω,

(ψrjop

γjM (h)ω

)∗= ωrjop

−γj−n−jM (h)ψ,

(ψrjop

γjM (h)ϕ

)∗= ϕrjop

−γj−n−jM (h)ψ,

with h(z) = h(n + 1 − j − z)(∗), and from the already proven result we finally

obtain assertion a). Note that if h ∈ Mµ;ℓV,P (X ; H 1

2−γj;E,F ) then ωrjop

γjM (h)ϕ,

ψrjopγjM (h)ω and ψrjop

γjM (h)ϕ are Volterra operators that generate asymptotics

since they fulfill the defining mapping property in Definition 6.2.1, which followsfrom Proposition 5.3.5.

Let us now prove b). Due to Lemma 6.2.13 we may write

ωrjopγjM (h)ω = ωrjop

γ−n2 −ε

M (h)ω +G


with a Green operator G ∈ CG(X∧, (γ, (−∞, 0]);E,F ) which is independent ofε > 0, provided that ε > 0 is sufficiently small. Consequently, the operatorωrjop

γjM (h)ω −G is continuous in the spaces

Sγ(X∧, E) −→⋂

ε>0

Sγ+j−ε(X∧, F ) = Sγ(−j,0](X∧, F ),

K(s,t),γ;ℓ(X∧, E)δ −→⋂

ε>0

K(s−µ,t),γ+j−ε;ℓ(X∧, F )∞ = K(s−µ,t),γ;ℓ(−j,0] (X∧, F )∞,

which shows that there is an asymptotic type Q ∈ As((γ, (−j, 0]), C∞(X,F )

)such

that

ωrjopγjM (h)ω :

Sγ(X∧, E) −→ SγQ(X∧, F ),

K(s,t),γ;ℓ(X∧, E)δ −→ K(s−µ,t),γ;ℓQ (X∧, F )∞

for all s, t, δ ∈ R. From Theorem 6.2.12 we conclude that the same arguments apply

to the formal adjoint operator. If h ∈Mµ;ℓV,P (X ; H 1

2−γj;E,F ) then ωrjop

γjM (h)ω is a

Volterra operator that generates asymptotics since it fulfills the defining mappingproperty in Definition 6.2.1.

Lemma 6.2.16. Let ωj, ωj , ωj, ωj ∈ C∞0 (R+) be cut-off functions. Moreover, let

γj , γj ∈ R such that γ − n2 − j ≤ γj , γj ≤ γ − n

2 , and let hj ∈ Mµ;ℓPj

(X ;E,F ) with

πCPj∩Γ 12−γj

= πCPj∩Γ 12−γj

= ∅. Then

N−1∑

j=0

ωjrjop

γjM (hj)ωj −

N−1∑

j=0

ωjrjop

γjM (hj)ωj ∈ Cµ;ℓ

G (X∧, (γ, (−∞, 0]);E,F ).

If hj ∈Mµ;ℓV,Pj

(X ; H 12−γj

;E,F )∩Mµ;ℓV,Pj

(X ; H 12−γj

;E,F ) then

N−1∑

j=0

ωjrjop

γjM (hj)ωj −

N−1∑

j=0

ωjrjop

γjM (hj)ωj ∈ Cµ;ℓ

G,V (X∧, (γ, (−∞, 0]);E,F ).

Proof. We have ωj = ωj + ψj and ωj = ωj + ϕj with ψj , ϕj ∈ C∞0 (R+). Thus the

assertion follows from Lemma 6.2.13 and Lemma 6.2.15.

Lemma 6.2.17. Let H be another vector bundle over X, and let ω, ω, ω, ω ∈C∞

0 (R+) be cut-off functions. Moreover, let γ1, γ2 ∈ R such that γ− n2 − k ≤ γ1 ≤

γ− n2 and γ− n

2 −j ≤ γ2 ≤ γ− n2 , and let g ∈Mµ;ℓ

P (cl)(X ;F,H), h ∈Mµ′;ℓQ(cl)(X ;E,F )

with πCP∩Γ 12−γ1

= ∅, πCQ∩Γ 12−γ2

= ∅. Then

(ωrkopγ1M (g)ω

)(ωrjopγ2M (h)ω

)= ωrk+jopγM ((T−jg)h)ω +G

with an operator G ∈ Cµ+µ′;ℓG (X∧, (γ, (−∞, 0]);E,H), and γ − n

2 − k − j ≤ γ ≤γ− n

2 . Here T denotes the translation operator for functions in the complex plane,i. e., (

(T−jg)h)(z) = g(z − j)h(z) ∈Mµ+µ′;ℓ

R(cl) (X ;E,H).


If even g ∈ Mµ;ℓV,P (cl)(X ; H 1

2−γ1;F,H) and h ∈ Mµ′;ℓ

V,Q(cl)(X ; H 12−γ2

;E,F ) we

may also choose γ ∈ R such that (T−jg)h ∈ Mµ+µ′;ℓV,R(cl)(X ; H 1

2−γ;E,H), and

G ∈ Cµ+µ′;ℓG,V (X∧, (γ, (−∞, 0]);E,H).

Proof. We may write(ωrkopγ1M (g)ω

)(ωrjopγ2M (h)ω

)= rk+j

(ωopγ1−jM (T−jg)ω

)(ωopγ2M (h)ω

).

Choose γ1 − j ≤ γ ≤ γ2 such that no singularity of T−jg and h lies on the weightline Γ 1

2−γ. From Lemma 6.2.13 we conclude

(ωopγ1−jM (T−jg)ω

)=

(ωopγM (T−jg)ω

)+G1,

(ωopγ2M (h)ω

)=

(ωopγM (h)ω

)+G2,

with Green operators

G1 ∈ CG(X∧, (γ +n

2, γ1 − j +

n

2, (−∞, 0]);F,H),

G2 ∈ CG(X∧, (γ2 +n

2, γ +

n

2, (−∞, 0]);E,F ).

Using Proposition 6.2.11 we obtain the following:

• G1G2 ∈ CG(X∧, (γ2 + n2 , γ1 − j + n

2 , (−∞, 0]);E,H),

•(ωopγM (T−jg)ω

)G2 ∈ CG(X∧, (γ2 + n

2 , γ + n2 , (−∞, 0]);E,H),

• G1

(ωopγM (h)ω

)∈ CG(X∧, (γ + n

2 , γ1 − j + n2 , (−∞, 0]);E,H),

and consequently

rk+j(ωopγ1−jM (T−jg)ω

)(ωopγ2M (h)ω

)≡ rk+j

(ωopγM (T−jg)ω

)(ωopγM (h)ω

)

modulo CG(X∧, (γ, (−∞, 0]);E,H). We may write(ωrk+jopγM (T−jg)ω

)(ωopγM (h)ω

)=

(ωrk+jopγM ((T−jg)h)ω

)

−(ωrk+jopγM (T−jg)(1 − ωω)opγM (h)ω

)︸︷︷︸

=: G

.

According to Proposition 6.2.11 we have for s, t, δ ∈ R

G :K(s,t),γ;ℓ(X∧, E)δ → K(s,t),γ+n2 ;ℓ(X∧, E)δ −→

(opγM(h)ω)H(s−µ′,t),γ+n

2 ;ℓ(X∧, F )

−→(1−ωω)

H(s−µ′,t),γ+n2 ;ℓ

(−∞,0] (X∧, F ) −→(opγM (T−jg))

H(s−µ−µ′,t),γ+n2 ;ℓ

Q (X∧, H)

−→(ωrk+j)

K(s−µ−µ′,t),γ;ℓ

Q(X∧, H)∞

with certain asymptotic types Q, Q.Analogously, we obtain G : Sγ(X∧, E) −→ Sγ

Q(X∧, H), and the same ar-

guments also apply to the formal adjoint operator G∗, i. e., we have G ∈Cµ+µ′;ℓG (X∧, (γ, (−∞, 0]);E,H). This implies the first assertion.


In case of Volterra operators we first observe that due to Lemma 6.2.13 wemay choose the weight γ := γ − n

2 − k − j, which produces a Green operator aserror term, i. e.,

G :=(ωrkopγ1M (g)ω

)(ωrjopγ2M (h)ω

)− ωrk+jop

γ−n2 −k−j

M ((T−jg)h)ω

∈ Cµ+µ′ ;ℓG (X∧, (γ, (−∞, 0]);E,H).

We have (T−jg)h ∈ Mµ+µ′;ℓV,R(cl)(X ; Hn+1

2 −γ+k+j ;E,H), and consequently G fulfills

the defining mapping property for Volterra operators that generate asymptoticsin Definition 6.2.1, which follows from Proposition 5.3.5. This proves the lemma.

Proposition 6.2.18. Let ωj , ωj ∈ C∞0 (R+) be cut-off functions near r = 0. More-

over, let hj ∈ Mµ;ℓPj


= ∅ with γ − n2 − j ≤ γj ≤

γ − n2 . Assume that

N−1∑

j=0

ωjrjop

γjM (hj)ωj : Sγ(X∧, E) −→ SγQ(X∧, F )

with an asymptotic type Q ∈ As((γ, (θ, 0]), C∞(X,F )

), where −∞ ≤ θ < −(N−1).

Then hj ≡ 0 for j = 0, . . . , N − 1.

Proof. The proof follows by induction over N ∈ N: Let N = 1. From Proposition6.2.11 we conclude

(1 − ω0)opγ−n

2

M (h0) : Tγ−n2(X∧, E) −→ Tγ−n

2 ,(−∞,0](X∧, F ),

ω0opγ−n

2

M (h0)(1 − ω0) : Tγ−n2(X∧, E) −→ Tγ−n

2 ,Q(X∧, F )

with an asymptotic type Q ∈ As((γ, (−∞, 0]), C∞(X,F )

). Using the assumption

we obtain

opγ−n

2

M (h0) : Tγ−n2(X∧, E) −→ Tγ−n

2,R(X∧, F )

with an asymptotic type R ∈ As((γ, (θ, 0]), C∞(X,F )

), and by possibly passing

to a smaller weight interval (θ, 0] we may assume that R = (θ, 0] is the emptyasymptotic type. Consequently, h0 acts as a multiplier in the spaces

h0 : S(Γn+12 −γ , C

∞(X,E)) −→ Mγ−n2

(Tγ−n

2 ,(θ,0](X∧, F )

).

Let ϕ ∈ C∞0 (Γn+1

2 −γ) such that ϕ ≡ 0 for |Im(z)| > 2, and ϕ ≡ 1 for

|Im(z)| ≤ 1. Hence ϕ(z)(h0(z)u) ≡ 0 for z ∈ Γn+12 −γ such that |Im(z)| > 2,

for all u ∈ C∞(X,E). From Theorem 4.2.16 and uniqueness of analytic continu-ation we obtain ϕ(z)(h0(z)u) ≡ 0 for all z ∈ Γn+1

2 −γ and all u ∈ C∞(X,E), and

thus h0(z) = 0 for all z ∈ Γn+12 −γ such that |Im(z)| ≤ 1. By the meromorphy of

h0 we conclude h0 ≡ 0 everywhere on C. This finishes the proof in the case N = 1.


Assume we have already proven the proposition for some N ∈ N, and

N∑

j=0

ωjrjop

γjM (hj)ωj : Sγ(X∧, E) −→ SγQ(X∧, F )

with an asymptotic type Q ∈ As((γ, (θ, 0]), C∞(X,F )

), where −∞ ≤ θ < −N .

By Lemma 6.2.15 the operator ωNrNopγNM (hN )ωN ∈ Cµ;ℓ

G (X∧, (γ, (−N, 0]);E,F ),which shows that

N−1∑

j=0

ωjrjop

γjM (hj)ωj : Sγ(X∧, E) −→ Sγ

Q(X∧, F )

with an asymptotic type Q ∈ As((γ, (−N, 0]), C∞(X,F )

).

Hence hj ≡ 0 for j = 0, . . . , N − 1 by induction, i. e.,

ωNrNopγNM (hN )ωN : Sγ(X∧, E) −→ SγQ(X∧, F ),

and consequently

ωNopγNM (hN )ωN : Sγ(X∧, E) −→ Sγ−NR (X∧, F )

with an asymptotic type R ∈ As((γ −N, (θ, 0]), C∞(X,F )

).

Choose γ − n2 ≤ γ < γ − n

2 − N − θ such that πCPN∩Γ 12−γ

= ∅. Due to

Lemma 6.2.13 we may write

ωNopγM (hN )ωN = ωNopγNM (hN )ωN +G

with a Green operator G ∈ CG(X∧, (γ + n2 , γN + n

2 , (−∞, 0]);E,F ). This shows

ωNopγM (hN )ωN : S γ+n2 (X∧, E) −→ Sγ−NR′ (X∧, F )∩S γ+n

2 (X∧, F )

with an asymptotic type R′ ∈ As((γ −N, (θ, 0]), C∞(X,F )

), while

Sγ−NR′ (X∧, F )∩S γ+n2 (X∧, F ) → S γ+n

2

(θ,0](X∧, F )

with some −∞ ≤ θ < 0. Hence the first part of the proof implies hN ≡ 0, and byinduction the proposition is proved.

Remark 6.2.19. Let A ∈ Cµ,;ℓM+G(cl)(X∧, (γ, (−N, 0]);E,F ). From Lemma 6.2.16

we obtain that in the representation (6.2.5) any change of the cut-off functionsωj , ωj as well as of the weights γ − n

2 − j ≤ γj ≤ γ − n2 results in an error in

Cµ;ℓG (X∧, (γ, (−∞, 0]);E,F ) only.

Consequently, the following simpler representation of conormal operators isvalid:

An operator A : Sγ(X∧, E) −→ Sγ(X∧, F ) is a conormal operator in the

space Cµ,;ℓM+G(cl)(X∧, (γ, (−N, 0]);E,F ) if and only if

A =

N−1∑

j=0

ωrjopγjM (hj)ω +G (6.2.6)


with an operator G ∈ Cµ,;ℓG (X∧, (γ, (−N, 0]);E,F ), and

i) a cut-off function ω ∈ C∞0 (R+) near r = 0,

ii) γ − n2 − j ≤ γj ≤ γ − n

2 ,

iii) meromorphic Mellin symbols hj ∈Mµ;ℓPj(cl)


=

∅.Moreover, A : Sγ(X∧, E) −→ Sγ(X∧, F ) is a Volterra conormal operator in the

space Cµ,;ℓM+G,V (cl)(X∧, (γ, (−N, 0]);E,F ) if and only if

A =

N−1∑

j=0

ωrjopγ−n

2 −j

M (hj)ω +G (6.2.7)

with a Volterra operator G ∈ Cµ,;ℓG,V (X∧, (γ, (−N, 0]);E,F ), and


ii) meromorphic Volterra Mellin symbols hj ∈Mµ;ℓV,Pj(cl)

(X ; Hn+12 −γ+j ;E,F ).

Definition 6.2.20. We define the conormal symbol mapping

σM : Cµ,;ℓM+G(,V (cl))(X∧, (γ, (−N, 0]);E,F ) −→ Σµ;ℓ

M(,V (cl))(X, (γ, (−N, 0]);E,F )

as follows:Let A ∈ Cµ,;ℓM+G(,V (cl))(X

∧, (γ, (−N, 0]);E,F ), and let

A =

N−1∑

j=0

ωjrjop

γjM (hj)ωj +G

be a representation according to (6.2.5). Then σM (A) is defined as

σM (A) := (σ0M (A), . . . , σ

−(N−1)M (A)) := (h0, . . . , hN−1). (6.2.8)

The component σ−kM (A) is called the conormal symbol of order −k of the operator

A. The conormal symbol σ0M (A) of order 0 is also called the conormal symbol

simply.

Theorem 6.2.21. a) Cµ,;ℓM+G(,V (cl))(X∧, (γ, (−N, 0]);E,F ) is a linear space.

b) The conormal symbol mapping is well-defined, and provides a linear surjection

σM : Cµ,;ℓM+G(,V (cl))(X∧, (γ, (−N, 0]);E,F ) → Σµ;ℓ

M(,V (cl))(X, (γ, (−N, 0]);E,F )

with kernel

ker(σM ) = Cµ,;ℓG(,V )(X∧, (γ, (−N, 0]);E,F ).

c) The quotient spaces

Quotµ;ℓM+G(,V (cl))(X

∧, (γ, (−N, 0]);E,F ) :=

Cµ,;ℓM+G(,V (cl))(X∧, (γ, (−N, 0]);E,F )/Cµ,;ℓG(,V )(X

∧, (γ, (−N, 0]);E,F )


do not depend on ∈ R, and for µ′ ≥ µ the embedding

Quotµ;ℓM+G(,V (cl))(X

∧, (γ, (−N, 0]);E,F ) →

Quotµ′;ℓM+G(,V (cl))(X

∧, (γ, (−N, 0]);E,F )

is well-defined.d) Taking the formal adjoint ∗ with respect to the r−

n2 L2-inner product induces

antilinear mappings

Cµ,;ℓM+G(cl)(X∧, (γ, (−N, 0]);E,F ) −→ Cµ,;ℓM+G(cl)(X

∧, (−γ, (−N, 0]);F,E),

Quotµ;ℓM+G(cl)(X

∧, (γ, (−N, 0]);E,F ) −→ Quotµ;ℓM+G(cl)(X

∧, (−γ, (−N, 0]);F,E).

For A ∈ Cµ,;ℓM+G(cl)(X∧, (γ, (−N, 0]);E,F ) we have σM (A∗) =

(σM (A)

)∗with

the ∗-operation (6.2.4).e) Let H be another vector bundle over X. The composition as operators on

Sγ(X∧, E) is well-defined in the spaces

Cµ,;ℓM+G(,V (cl))(X∧, (γ, (−N, 0]);F,H)×Cµ

′,′;ℓM+G(,V (cl))(X

∧, (γ, (−N, 0]);E,F )

−→ Cµ+µ′,+′;ℓM+G(,V (cl))(X

∧, (γ, (−N, 0]);E,H).

For

A ∈Cµ,;ℓM+G(,V (cl))(X∧, (γ, (−N, 0]);F,H),

B ∈Cµ′,′;ℓ

M+G(,V (cl))(X∧, (γ, (−N, 0]);E,F )

we have σM (AB) = σM (A)#σM (B) with the Mellin translation product (6.2.3).In particular,

Cµ,;ℓG(,V )(X∧, (γ, (−N, 0]);F,H)×Cµ

′,′;ℓM+G(,V )(X

∧, (γ, (−N, 0]);E,F )

−→ Cµ+µ′,+′;ℓG(,V ) (X∧, (γ, (−N, 0]);E,H),

Cµ,;ℓM+G(,V )(X∧, (γ, (−N, 0]);F,H)×Cµ

′,′;ℓG(,V ) (X∧, (γ, (−N, 0]);E,F )

−→ Cµ+µ′,+′;ℓG(,V ) (X∧, (γ, (−N, 0]);E,H),

and the composition is well-defined on the quotient spaces.

Proof. Let A,B ∈ Cµ,;ℓM+G(,V (cl))(X∧, (γ, (−N, 0]);E,F ). According to (6.2.6),

(6.2.7) we may write

A =N−1∑

j=0

ωrjopγjM (hj)ω +G, B =

N−1∑

j=0

ωrjopγjM (hj)ω + G,

and thus

λ1A+ λ2B =

N−1∑

j=0

ωrjopγjM (λ1hj + λ2hj)ω + (λ1G+ λ2G)


for λ1, λ2 ∈ C. This proves a). b) follows from a), Lemma 6.2.16 and Proposition6.2.18, and c) is a consequence of b). d) is subject to Theorem 6.2.12. It remainsto prove e). Note first that by Theorem 6.2.12 the composition is well-defined inthe spaces

Cµ,;ℓG(,V )(X∧, (γ, (−N, 0]);F,H)×Cµ

′,′;ℓM+G(,V )(X

∧, (γ, (−N, 0]);E,F )

−→ Cµ+µ′,+′;ℓG(,V ) (X∧, (γ, (−N, 0]);E,H),

Cµ,;ℓM+G(,V )(X∧, (γ, (−N, 0]);F,H)×Cµ

′,′;ℓG(,V ) (X∧, (γ, (−N, 0]);E,F )

−→ Cµ+µ′,+′;ℓG(,V ) (X∧, (γ, (−N, 0]);E,H),

and consequently the complete assertion e) follows from Lemma 6.2.15 and Lemma6.2.17.

Remark 6.2.22. By Theorem 6.2.21 we have the following:

The conormal operators Cµ,;ℓM+G(cl)(X∧, (γ, (−N, 0]);E,F ) form a (bi-)

graded ∗-algebra, and the conormal symbol mapping induces a ∗-homomorphismof graded algebras onto the algebra of conormal symbols. The kernel of this ho-

momorphism is the (bi-)graded symmetric ideal Cµ,;ℓG (X∧, (γ, (−N, 0]);E,F )of operators that generate asymptotics.

The Volterra conormal operators Cµ,;ℓM+G,V (cl)(X∧, (γ, (−N, 0]);E,F ) are a

(bi-)graded subalgebra, and the conormal symbol mapping restricts to a homo-morphism of graded algebras onto the algebra of Volterra conormal symbols. The

kernel of the restriction is the (bi-)graded ideal Cµ,;ℓG,V (X∧, (γ, (−N, 0]);E,F ) ofVolterra operators that generate asymptotics.

Smoothing Mellin and Green operators.

Definition 6.2.23. We define the space of smoothing (Volterra) Mellin and Greenoperators with respect to the weight datum (γ, (−N, 0]), N ∈ N, as

CM+G(,V )(X∧, (γ, (−N, 0]);E,F ) :=

⋂

µ,∈R

Cµ,;ℓM+G(,V )(X∧, (γ, (−N, 0]);E,F ).

Consequently, A : Sγ(X∧, E) −→ Sγ(X∧, F ) is a smoothing Mellin and Greenoperator in CM+G(X∧, (γ, (−N, 0]);E,F ) if and only if

A =N−1∑

j=0

ωrjopγjM (hj)ω +G (6.2.9)

with a Green operator G ∈ CG(X∧, (γ, (−N, 0]);E,F ), and


ii) γ − n2 − j ≤ γj ≤ γ − n

2 ,

iii) meromorphic Mellin symbols hj ∈ M−∞Pj


=

∅.


Moreover, A : Sγ(X∧, E) −→ Sγ(X∧, F ) is a smoothing Volterra Mellin andGreen operator in CM+G,V (X∧, (γ, (−N, 0]);E,F ) if and only if

A =

N−1∑

j=0

ωrjopγ−n

2 −j

M (hj)ω +G (6.2.10)

with a Volterra Green operator G ∈ CG,V (X∧, (γ, (−N, 0]);E,F ), and


ii) meromorphic Volterra Mellin symbols hj ∈M−∞V,Pj

(X ; Hn+12 −γ+j ;E,F ).

Remark 6.2.24. According to Theorem 6.2.21 the smoothing (Volterra) Mellin andGreen operators form an ideal in the algebra of (Volterra) conormal operators, i. e.,the composition as operators on Sγ(X∧, E) is well-defined in the spaces

Cµ,;ℓM+G(,V )(X∧, (γ, (−N, 0]);F,H)×CM+G(,V )(X

∧, (γ, (−N, 0]);E,F )

−→ CM+G(,V )(X∧, (γ, (−N, 0]);E,H),

CM+G(,V )(X∧, (γ, (−N, 0]);F,H)×Cµ,;ℓM+G(,V )(X

∧, (γ, (−N, 0]);E,F )

−→ CM+G(,V )(X∧, (γ, (−N, 0]);E,H)

for vector bundles E,F,H ∈ Vect(X).

Definition 6.2.25. a) Let A ∈ CM+G(X∧, (γ, (−N, 0]);E). Then the operator1 + A is called elliptic if there exists s0 ∈ R such that the operator family1 + σ0

M (A)(z) : Hs0(X,E) −→ Hs0(X,E) is bijective for all z ∈ Γn+12 −γ .

b) Let A ∈ CM+G,V (X∧, (γ, (−N, 0]);E). The operator 1 + A is called parabolicif there exists s0 ∈ R such that 1 + σ0

M (A)(z) : Hs0(X,E) −→ Hs0(X,E) isbijective for all z ∈ Hn+1

2 −γ .

Remark 6.2.26. The identity belongs to C0,0;ℓM+G,V cl(X

∧, (γ, (−N, 0]);E) with co-

normal symbol given as σM (1) = 1:With a cut-off function ω ∈ C∞

0 (R+) near r = 0 write

1 = ω1ω +(ω1(1 − ω) + (1 − ω)1

),

where(ω1(1 − ω) + (1 − ω)1

)∈ C0,0;ℓ

G,V (X∧, (γ, (−∞, 0]);E).

Let A ∈ CM+G(,V )(X∧, (γ, (−N, 0]);E). Then we see that 1 + A is elliptic

(parabolic) in the sense of Definition 6.2.25 if and only if σM (1 +A) = 1 + σM (A)is elliptic (parabolic) in the sense of Definition 6.2.5.

Theorem 6.2.27. a) Let A ∈ CM+G(X∧, (γ, (−N, 0]);E). Then the following areequivalent:

i) 1 +A is elliptic in the sense of Definition 6.2.25.ii) There exists B ∈ CM+G(X∧, (γ, (−N, 0]);E) such that (1 + A)(1 + B) =

1 + G1 and (1 + B)(1 + A) = 1 + G2 with Green operators G1, G2 ∈CG(X∧, (γ, (−N, 0]);E).

b) Let A ∈ CM+G,V (X∧, (γ, (−N, 0]);E). Then the following are equivalent:


i) 1 +A is parabolic in the sense of Defintion 6.2.25.ii) There exists B ∈ CM+G,V (X∧, (γ, (−N, 0]);E) such that (1+A)(1+B) = 1

and (1 +B)(1 +A) = 1, i. e., 1 +A is invertible with inverse (1 +A)−1 =1 +B.

Proof. According to Corollary 6.2.8 the operator 1 + A is elliptic (parabolic)if and only if there exists a (Volterra) conormal symbol b := (g0, . . . , gN−1) ∈ΣM(,V )(X

∧, (γ, (−N, 0]);E) such that 1 + σM (A) is invertible with respect to theMellin translation product with inverse 1 + b. Hence we conclude from Theorem6.2.21 that the conditions ii) in a) and b) are sufficient for the ellipticity (parabol-icity) of the operator 1 +A.

Now assume that 1 + A is elliptic (parabolic). With (g0, . . . , gN−1) we asso-ciate an operator C ∈ CM+G(,V )(X

∧, (γ, (−N, 0]);E) via

C =

N−1∑

j=0

ωrjopγjM (gj)ω

in the sense of (6.2.9) or (6.2.10), respectively. Theorem 6.2.21 implies (1+A)(1+C) = 1+G1 and (1+C)(1+A) = 1+G2 with (Volterra) Green operators G1, G2 ∈CG(,V )(X

∧, (γ, (−N, 0]);E). Hence the proof of a) is finished with B := C.In case of b) the operatorsG1 and G2 are Volterra Green operators. Hence, by

Theorem 6.1.6, 1+G1 and 1+G2 are invertible with inverses (1+G1)−1 = 1+ G1

and (1+G2)−1 = 1+G2, where G1, G2 ∈ CG,V (X∧, (γ, (−N, 0]);E). Consequently,

1 +A is invertible with inverse (1 +A)−1 = 1 +B, where

B := G1 + C + CG1 = G2 + C + G2C ∈ CM+G,V (X∧, (γ, (−N, 0]);E).


6.3. The algebra of Volterra cone operators

Notation 6.3.1. Let Y be a topological space. For functions ϕ, ψ : Y −→ C wewrite ϕ ≺ ψ if ψ ≡ 1 in a neighbourhood of supp(ϕ).

Definition 6.3.2. Let (γ, (−N, 0]) be a weight datum, N ∈ N, and let µ, ∈ R.

a) We define the space Cµ,;ℓ(cl) (X∧, (γ, (−N, 0]);E,F ) of (classical) cone pseu-

dodifferential operators (of order (µ, )) associated with the weight datum(γ, (−N, 0]) as follows:

A : Sγ(X∧, E) −→ Sγ(X∧, F ) belongs to Cµ,;ℓ(cl) (X∧, (γ, (−N, 0]);E,F ) if

and only if• for all cut-off functions ω, ω ∈ C∞

0 (R+) near r = 0 we have

ωAω = opγ−n

2

M (h) +AM+G (6.3.1)

with some h ∈ C∞B (R+,M

µ;ℓO(cl)(X ;E,F )), and a smoothing Mellin and

Green operator AM+G ∈ CM+G(X∧, (γ, (−N, 0]);E,F ),


• for all cut-off functions ω, ω ∈ C∞0 (R+) near r = 0 we may write

(1 − ω)A(1 − ω) = opr(a) (6.3.2)

with some a ∈ S(R, Lµ;ℓ(cl)(X ; R;E,F )),

• for all cut-off functions ω, ω ∈ C∞0 (R+) near r = 0 such that ω ≺ ω we

have

ωA(1 − ω), (1 − ω)Aω ∈ CG(X∧, (γ, (−N, 0]);E,F ). (6.3.3)

b) The subspace Cµ,;ℓV (cl)(X∧, (γ, (−N, 0]);E,F ) of (classical) Volterra cone pseu-

dodifferential operators (of order (µ, )) associated with the weight datum(γ, (−N, 0]) is defined as follows:

A : Sγ(X∧, E) −→ Sγ(X∧, F ) belongs to Cµ,;ℓV (cl)(X∧, (γ, (−N, 0]);E,F ) if

and only if• for all cut-off functions ω, ω ∈ C∞

0 (R+) near r = 0 we have

ωAω = opγ−n

2

M (h) +AM+G (6.3.4)

with some h ∈ C∞B (R+,M

µ;ℓV,O(cl)(X ;E,F )), and a smoothing Volterra

Mellin and Green operator AM+G ∈ CM+G,V (X∧, (γ, (−N, 0]);E,F ),

• for all cut-off functions ω, ω ∈ C∞0 (R+) near r = 0 we may write

(1 − ω)A(1 − ω) = opr(a) (6.3.5)

with some a ∈ S(R, Lµ;ℓV (cl)(X ; H;E,F )),

• for all cut-off functions ω, ω ∈ C∞0 (R+) near r = 0 such that ω ≺ ω we

have

ωA(1 − ω), (1 − ω)Aω ∈ CG,V (X∧, (γ, (−N, 0]);E,F ). (6.3.6)

Theorem 6.3.3. An operator A : C∞0 (X∧, E) −→ C∞(X∧, F ) belongs to

Cµ,;ℓ(V (cl))(X∧, (γ, (−N, 0]);E,F ) if and only if for some (all) cut-off functions

ω3 ≺ ω1 ≺ ω2 we may write

A = ω1opγ−n

2

M (h)ω2 + (1 − ω1)opr(a)(1 − ω3) +AM+G, (6.3.7)

where

AM+G ∈ CM+G(,V )(X∧, (γ, (−N, 0]);E,F ),

h ∈ C∞(R+,Mµ;ℓ(V,)O(cl)(X ;E,F )),

a ∈S(R, Lµ;ℓ

(cl)(X ; R;E,F ))


Proof. Let A ∈ Cµ,;ℓ(V (cl))(X∧, (γ, (−N, 0]);E,F ), and let ω3 ≺ ω1 ≺ ω2 be arbitrary

cut-off functions near r = 0. Moreover, let ω, ω ∈ C∞0 (R+) be cut-off functions

such that ω ≺ ωj ≺ ω for j = 1, 2, 3. We write

A = ω1

(ωAω

)ω2+(1−ω1)

((1−ω)A(1−ω)

)(1−ω3)+

(ω1A(1−ω2)+(1−ω1)Aω3

),


and consequently A is of the form (6.3.7) by Definition 6.3.2.For the proof of the converse note that it suffices to treat each term in the

representation (6.3.7) separately:

Step 1: A = ω1opγ−n

2

M (h)ω2 ∈ Cµ,;ℓ(V (cl))(X∧, (γ, (−N, 0]);E,F ):

Let ω, ω ∈ C∞0 (R+) be arbitrary cut-off functions near r = 0. We have

ωAω = opγ−n

2

M (ω(r)ω1(r)h(r, z)ω2(r′)ω(r′)),

ω(r)ω1(r)h(r, z)ω2(r′)ω(r′) ∈ C∞

B (R+×R+,Mµ;ℓ(V,)O(cl)(X ;E,F )),

and thus we conclude from Theorem 5.3.2 that ωAω = opγ−n

2

M (g) with the

left-symbol g ∈ C∞B (R+,M

µ;ℓ(V,)O(cl)(X ;E,F )) associated with the double-symbol

ω(r)ω1(r)h(r, z)ω2(r′)ω(r′). This proves that ωAω is of the form (6.3.1) or (6.3.4),

respectively.Next consider

(1 − ω)A(1 − ω) =((1 − ω)ω1

)opγ−n

2

M (h)(ω2(1 − ω)

)

= ψ1opγ−n

2

M (h)ψ2

with ψ1, ψ2 ∈ C∞0 (R+). According to Theorem 5.2.10 and Theorem 2.6.18 we may

write

opγ−n

2

M (h) = opr(a) + opγ−n

2

M ((1 − ϕ)(r′r

)h)

with a function ϕ ∈ C∞0 (R+) such that ϕ ≡ 1 near r = 1, and a(r, τ) :=

Qγ−n2(ϕ, h)(r, rτ), where Qγ−n

2is the inverse Mellin quantization with respect

to the weight γ − n2 (cf. Definition 2.6.15). Moreover, Theorem 5.2.10 implies

ψ1(r)a(r, τ)ψ2(r′) ∈

S−∞,−∞(R×R, Lµ;ℓ

(cl)(X ; R;E,F ))

S−∞,−∞(R×R, Lµ;ℓV (cl)(X ; H;E,F )),

and from Theorem 5.4.3 we conclude that ψ1opr(a)ψ2 = opr(a) with

a ∈S−∞(R, Lµ;ℓ

(cl)(X ; R;E,F ))

S−∞(R, Lµ;ℓV (cl)(X ; H;E,F )).

Next observe that

ψ1opγ−n

2

M ((1 − ϕ)(r′r

)h)ψ2 = op

γ−n2

M (ψ1(r)(1 − ϕ)(r′r

)ψ2(r

′)h),

ψ1(r)(1 − ϕ)(r′r

)ψ2(r

′)h(r, z) ∈ C∞B (R+×R+,M

µ;ℓ(V,)O(cl)(X ;E,F )),

ψ1(r)(1 − ϕ)(r′r

)ψ2(r

′)h(r, z) ≡ 0 for | rr′

− 1| < ε

with a sufficiently small ε > 0. Proposition 5.3.4 implies

ψ1opγ−n

2

M ((1 − ϕ)(r′r

)h)ψ2 = op

γ−n2

M (h)


with h ∈ C∞B (R+,M

−∞(V,)O(X ;E,F )). Let ψ ∈ C∞

0 (R+) such that ψ1, ψ2 ≺ ψ. Then

we see by construction that

opγ−n

2

M (h) = ψopγ−n

2

M (h)ψ,

and thus Theorem 5.3.6, Theorem 5.3.3 and Proposition 5.3.5 (in case of Volterraoperators) imply

opγ−n

2

M (h) ∈ CG(,V )(X∧, (γ, (−∞, 0]);E,F ).

So far we have proved that

(1 − ω)A(1 − ω) = opr(a) +G

with a as above, and G ∈ CG(,V )(X∧, (γ, (−∞, 0]);E,F ) such that G = ψGψ.

From Proposition 6.1.5 we obtain G = opr(b) with

b ∈S−∞(R, L−∞(X ; R;E,F ))

S−∞(R, L−∞V (X ; H;E,F )),

and consequently (1 − ω)A(1 − ω) = opr(c) with

c = a+ b ∈S−∞(R, Lµ;ℓ

(cl)(X ; R;E,F ))

S−∞(R, Lµ;ℓV (cl)(X ; H;E,F )).

This shows that (1 − ω)A(1 − ω) is of the form (6.3.2) or (6.3.5), respectively.Next assume that ω ≺ ω. Then we have

ωA(1 − ω) =(ωω1

)op

γ−n2

M (h)(ω2(1 − ω)

)

= ωopγ−n

2

M (h)ψ,

(1 − ω)Aω =((1 − ω)ω1

)op

γ−n2

M (h)(ω2ω

)

= ϕopγ−n

2

M (h)ω

with cut-off functions ω, ω ∈ C∞0 (R+) and ϕ, ψ ∈ C∞

0 (R+), such that ωψ ≡ 0 aswell as ωϕ ≡ 0. From Proposition 5.3.4 we conclude that

ωopγ−n

2

M (h)ψ = opγ−n

2

M (h1)

h1 ∈ C∞B (R+,M

−∞(V,)O(X ;E,F )),

ϕopγ−n

2

M (h)ω = opγ−n

2

M (h2)

h2 ∈ C∞B (R+,M

−∞(V,)O(X ;E,F )).

Let ψ ∈ C∞0 (R+) and η ∈ C∞

0 (R+) such that ψ, ϕ ≺ ψ as well as ω, ω ≺ η. Byconstruction we have

opγ−n

2

M (h1) = ηopγ−n

2

M (h1)ψ,

opγ−n

2

M (h2) = ψopγ−n

2

M (h2)η.


From Proposition 2.6.4 we conclude opγ−n

2

M (h1) = opγ′−n

2

M (h1) as operators onC∞

0 (X∧, E), for all γ′ ∈ R. Hence we obtain from Theorem 5.3.6

ηopγ−n

2

M (h1)ψ : K(s,t),γ;ℓ(X∧, E)δ −→ Sγ(−∞,0](X∧, F ),

ψopγ−n

2

M (h2)η : K(s,t),γ;ℓ(X∧, E)δ −→ Sγ(−∞,0](X∧, F )

for all s, t, δ ∈ R. Using Theorem 5.3.3 we obtain with the same reasoning(ηop

γ−n2

M (h1)ψ)∗

= ψop−γ−n

2

M (h∗1)η : K(s,t),−γ;ℓ(X∧, F )δ −→ S−γ(−∞,0](X

∧, E),(ψop

γ−n2

M (h2)η)∗

= ηop−γ−n

2

M (h∗2)ψ : K(s,t),−γ;ℓ(X∧, F )δ −→ S−γ(−∞,0](X

∧, E)

for all s, t, δ ∈ R. Summing up, using Proposition 5.3.5 in case of Volterra opera-tors, we have shown

ωA(1 − ω), (1 − ω)Aω ∈ CG(,V )(X∧, (γ, (−∞, 0]);E,F ).

This finishes the proof of Step 1.

Step 2: A = (1 − ω1)opr(a)(1 − ω3) ∈ Cµ,;ℓ(V (cl))(X∧, (γ, (−N, 0]);E,F ):

Let ω, ω ∈ C∞0 (R+) be arbitrary cut-off functions near r = 0. We have

(1 − ω)A(1 − ω) = opr((

(1 − ω(r))(1 − ω1(r)))a(r, τ)

((1 − ω3(r

′))(1 − ω(r′)))),

((1 − ω(r))(1 − ω1(r))

)a(r, τ)

((1 − ω3(r

′))(1 − ω(r′)))

∈S,0(R×R, Lµ;ℓ

(cl)(X ; R;E,F ))

S,0(R×R, Lµ;ℓV (cl)(X ; H;E,F )).

Consequently, (1 − ω)A(1 − ω) = opr(g), where

g ∈S(R, Lµ;ℓ

(cl)(X ; R;E,F ))

S(R, Lµ;ℓV (cl)(X ; H;E,F ))

is the left-symbol associated with the double-symbol((1 − ω(r))(1 − ω1(r))

)a(r, τ)

((1 − ω3(r

′))(1 − ω(r′)))

according to Theorem 5.4.3. This shows that (1−ω)A(1− ω) is of the form (6.3.2)or (6.3.5), respectively.

Next consider the operator

ωAω =(ω(1 − ω1)

)opr(a)

((1 − ω3)ω

)= ψ1opr(a)ψ2

with ψ1, ψ2 ∈ C∞0 (R+). According to Theorem 5.2.10 and Theorem 2.6.18 we may

write

opr(a) = opγ−n

2

M (h) + opr((1 − ϕ)

(r′r

)a),

where ϕ ∈ C∞0 (R+) such that ϕ ≡ 1 near r = 1, and h(r, z) := Q(ϕ, a)(r, z) with

a(r, τ) := a(r, r−1τ). Here Q denotes the Mellin quantization, see Definition 2.6.15.From Theorem 5.2.10 we obtain

ψ1(r)h(r, z)ψ2(r′) ∈ C∞

B (R+×R+,Mµ;ℓ(V,)O(cl)(X ;E,F )),


and consequently

ψ1opγ−n

2

M (h)ψ2 = opγ−n

2

M (h)

with the left-symbol h ∈ C∞B (R+,M

µ;ℓ(V,)O(cl)(X ;E,F )) associated with the double-

symbol ψ1(r)h(r, z)ψ2(r′) according to Theorem 5.3.2. Moreover, we have

ψ1(r)(1 − ϕ)(r′r

)a(r, τ)ψ2(r

′) ∈S−∞,−∞(R×R, Lµ;ℓ

(cl)(X ; R;E,F ))

S−∞,−∞(R×R, Lµ;ℓV (cl)(X ; H;E,F )),

ψ1(r)(1 − ϕ)(r′r

)a(r, τ)ψ2(r

′) ≡ 0 for |r − r′| < ε

with some sufficiently small ε > 0. Hence Proposition 5.4.5 implies

ψ1opr((1 − ϕ)

(r′r

)a)ψ2 = opr(c)

with a symbol

c ∈S−∞(R, L−∞(X ; R;E,F ))

S−∞(R, L−∞V (X ; H;E,F )).

Let ψ ∈ C∞0 (R+) such that ψ1, ψ2 ≺ ψ. Then we have ψopr(c)ψ = opr(c) by

construction, and from Theorem 5.4.7, Theorem 5.4.4 and Proposition 5.4.6 (incase of Volterra operators) we conclude

ψopr(c)ψ ∈ CG(,V )(X∧, (γ, (−∞, 0]);E,F ).

Summing up, we have shown that ωAω is of the form (6.3.1) or (6.3.4), respectively.Next assume that ω ≺ ω. Then we have

ωA(1 − ω) =(ω(1 − ω1)

)opr(a)

((1 − ω3)(1 − ω)

)

= ϕopr(a)(1 − ω),

(1 − ω)Aω =((1 − ω)(1 − ω1)

)opr(a)

((1 − ω3)ω

)

= (1 − ω)opr(a)ψ,

where ϕ, ψ ∈ C∞0 (R+), and ω, ω ∈ C∞

0 (R+) are cut-off functions satisfying ϕ ≺ ωand ψ ≺ ω. From Proposition 5.4.5 we conclude

ϕopr(a)(1 − ω) = opr(a1)

a1 ∈S−∞(R, L−∞(X ; R;E,F ))

S−∞(R, L−∞V (X ; H;E,F )),

(1 − ω)opr(a)ψ = opr(a2)

a2 ∈S−∞(R, L−∞(X ; R;E,F ))

S−∞(R, L−∞V (X ; H;E,F )).


Choose η ∈ C∞0 (R+) and ψ ∈ C∞

0 (R+) such that ϕ, ψ ≺ ψ, and η ≺ ω, ω. Thenwe have

ψopr(a1)(1 − η) = opr(a1),

(1 − η)opr(a2)ψ = opr(a2)

by construction. Theorem 5.4.7, Theorem 5.4.4 and Proposition 5.4.6 (in case ofVolterra operators) imply

ψopr(a1)(1 − η), (1 − η)opr(a2)ψ ∈ CG(,V )(X∧, (γ, (−∞, 0]);E,F ).

Summing up, we conclude

ωA(1 − ω), (1 − ω)Aω ∈ CG(,V )(X∧, (γ, (−∞, 0]);E,F ),

and the proof of Step 2 is finished.

Step 3: CM+G(,V )(X∧, (γ, (−N, 0]);E,F ) ⊆ Cµ,;ℓ(V (cl))(X

∧, (γ, (−N, 0]);E,F ):

Let A ∈ CM+G(,V )(X∧, (γ, (−N, 0]);E,F ), and let ω, ω ∈ C∞

0 (R+) be ar-bitrary cut-off functions. Then we have ωAω ∈ CM+G(,V )(X

∧, (γ, (−N, 0]);E,F )by definition of the smoothing (Volterra) Mellin and Green operators. Moreover,ωA(1 − ω), (1 − ω)Aω ∈ CG(,V )(X

∧, (γ, (−N, 0]);E,F ) by Lemma 6.2.15. Let

ω ∈ C∞0 (R+) be a cut-off function such that ω ≺ ω and ω ≺ ω. Note that

G := (1 − ω)A(1 − ω) ∈ CG(,V )(X∧, (γ, (−N, 0]);E,F ) due to Lemma 6.2.15, and

from the choice of ω we conclude G = (1 − ω)G(1 − ω). Hence G = opr(g) with

g ∈S−∞(R, L−∞(X ; R;E,F ))

S−∞(R, L−∞V (X ; H;E,F ))

due to Proposition 6.1.5, and the proof of Step 3 is complete.

Theorem 6.3.4. The following inclusions are valid:

CM+G(,V )(X∧, (γ, (−N, 0]);E,F ) ⊆ Cµ,;ℓ(V (cl))(X

∧, (γ, (−N, 0]);E,F ),

Cµ,;ℓ(V (cl))(X∧, (γ, (−N, 0]);E,F ) ⊆ Cµ,;ℓM+G(,V (cl))(X

∧, (γ, (−N, 0]);E,F ).

Let A ∈ Cµ,;ℓ(V (cl))(X∧, (γ, (−N, 0]);E,F ), and let

A = ω1opγ−n

2

M (h)ω2 + (1 − ω1)opr(a)(1 − ω3) +AM+G

be any representation of A according to (6.3.7). Then the conormal symbols of Aare given as

σ−kM (A)(z) =

1

k!(∂krh)(0, z) + σ−k

M (AM+G)(z)

for k = 0, . . . , N − 1.

Proof. The inclusion

CM+G(,V )(X∧, (γ, (−N, 0]);E,F ) ⊆ Cµ,;ℓ(V (cl))(X

∧, (γ, (−N, 0]);E,F )

is subject to Theorem 6.3.3.


Assume that A ∈ Cµ,;ℓ(V (cl))(X∧, (γ, (−N, 0]);E,F ) is given as

A = ω1opγ−n

2

M (h)ω2 + (1 − ω1)opr(a)(1 − ω3) +AM+G

in the sense of (6.3.7). For short, we set hk := 1k! (∂

kr h)(0, z) for k = 0, . . . , N − 1.

Then Taylor’s formula implies

A=(N−1∑

k=0

ω1rkop

γ−n2

M (hk)ω2+AM+G

)+

(ω1r

Nopγ−n

2

M (h)ω2+(1−ω1)opr(a)(1−ω3))

with a function h ∈ C∞B (R+,M

µ;ℓ(V,)O(cl)(X ;E,F )). Due to Theorem 5.4.7, Theorem

5.4.4 and Proposition 5.4.6 (in case of Volterra operators) we have

(1 − ω1)opr(a)(1 − ω3) ∈ Cµ,;ℓG(,V )(X∧, (γ, (−∞, 0]);E,F ).

Moreover, we conclude from Theorem 5.3.6

ω1rNop

γ−n2

M (h)ω2 :

K(s,t),γ;ℓ(X∧, E)δ −→ K(s−µ,t),γ+N ;ℓ(X∧, F )∞

Sγ(X∧, E) −→ Sγ+N (X∧, F ),

and by Theorem 5.3.3 and Proposition 2.6.4 we have(ω1r

Nopγ−n

2

M (h)ω2

)∗= ω2r

Nop−γ−n

2

M (T−N h∗)ω1,

and consequently

(ω1r

Nopγ−n

2

M (h)ω2

)∗:

K(s,t),−γ;ℓ(X∧, E)δ −→ K(s−µ,t),−γ+N ;ℓ(X∧, F )∞

S−γ(X∧, E) −→ S−γ+N(X∧, F ).

Using Proposition 5.3.5 (in case of Volterra operators) we thus obtain

ω1rNop

γ−n2

M (h)ω2 ∈ Cµ,;ℓG(,V )(X∧, (γ, (−N, 0]);E,F ).

This shows that

A ≡N−1∑

k=0

ω1rkop

γ−n2

M (hk)ω2 +AM+G

modulo Cµ,;ℓG(,V )(X∧, (γ, (−N, 0]);E,F ), and from Theorem 6.2.21 we obtain

A ∈ Cµ,;ℓM+G(,V (cl))(X∧, (γ, (−N, 0]);E,F ),

σ−kM (A) = hk + σ−k

M (AM+G)

for k = 0, . . . , N − 1 as asserted.

Corollary 6.3.5. Let A ∈ Cµ,;ℓ(X∧, (γ, (−N, 0]);E,F ). Then A extends by con-tinuity to an operator




Moreover, for every asymptotic type P ∈ As((γ, (−N, 0]), C∞(X,E)) thereexists an asymptotic type Q ∈ As((γ, (−N, 0]), C∞(X,F )) such that A restricts tocontinuous operators

A :

K(s,t),γ;ℓP (X∧, E)δ −→ K(s−µ,t),γ;ℓ

Q (X∧, F )δ−

SγP (X∧, E) −→ SγQ(X∧, F )


Let A ∈ Cµ,;ℓV (X∧, (γ, (−N, 0]);E,F ). Then A restricts for every r0 ∈ R+

to continuous operators

A :

Tγ−n

2,0((0, r0), C

∞(X,E)) −→ Tγ−n2,0((0, r0), C

∞(X,F ))

H(s,t),γ;ℓ0 ((0, r0]×X,E) −→ H(s−µ,t),γ;ℓ

0 ((0, r0]×X,F )

for all s, t ∈ R.

Proof. This follows from Theorem 6.3.4 and Theorem 6.2.12.

Corollary 6.3.6. For vector bundles E,F,H ∈ Vect(X) the composition as oper-ators on Sγ(X∧, E) is well-defined in the spaces

Cµ,;ℓ(V ) (X∧, (γ, (−N, 0]);F,H)×CG(,V )(X∧, (γ, (−N, 0]);E,F )

−→ CG(,V )(X∧, (γ, (−N, 0]);E,H),

CG(,V )(X∧, (γ, (−N, 0]);F,H)×Cµ,;ℓ(V ) (X∧, (γ, (−N, 0]);E,F )

−→ CG(,V )(X∧, (γ, (−N, 0]);E,H),

as well as

Cµ,;ℓ(V ) (X∧, (γ, (−N, 0]);F,H)×CM+G(,V )(X∧, (γ, (−N, 0]);E,F )

−→ CM+G(,V )(X∧, (γ, (−N, 0]);E,H),

CM+G(,V )(X∧, (γ, (−N, 0]);F,H)×Cµ,;ℓ(V ) (X∧, (γ, (−N, 0]);E,F )

−→ CM+G(,V )(X∧, (γ, (−N, 0]);E,H).

Proof. This follows from Theorem 6.3.4 and Theorem 6.2.21.

The symbolic structure.

Theorem 6.3.7. Let A ∈ Cµ,;ℓ(V (cl))(X∧, (γ, (−N, 0]);E,F ), and let 0 < T2 < T1 <

∞. There exist

h ∈ C∞B (R+,M

µ;ℓ(V,)O(cl)(X ;E,F )),

a ∈S(R, Lµ;ℓ

(cl)(X ; R;E,F ))


with the following properties:


For all cut-off functions ω, ω ∈ C∞0 (R+) near r = 0 such that χ[0,T2] ≺ ω, ω ≺

χ[0,T1] we have

ωAω − ωopγ−n

2

M (h)ω ∈ CM+G(,V )(X∧, (γ, (−N, 0]);E,F ),

(1 − ω)A(1 − ω) − (1 − ω)opr(a)(1 − ω) ∈ CG(,V )(X∧, (γ, (−N, 0]);E,F ).

(6.3.8)In particular, if

χ[0,T2] ≺ ω3 ≺ ω1 ≺ ω2 ≺ χ[0,T1] (6.3.9)

are cut-off functions, we have

A = ω1opγ−n

2

M (h)ω2 + (1 − ω1)opr(a)(1 − ω3) +AM+G (6.3.10)

with AM+G ∈ CM+G(,V )(X∧, (γ, (−N, 0]);E,F ).

Proof. Let ω, ω ∈ C∞0 (R+) be cut-off functions such that χ[0,T1] ≺ ω and ω ≺

χ[0,T2]. According to (6.3.1) and (6.3.4), respectively, we have

ωAω = opγ−n

2

M (h) + A,

h ∈ C∞B (R+,M

µ;ℓ(V,)O(cl)(X ;E,F )),

A ∈ CM+G(,V )(X∧, (γ, (−N, 0]);E,F ).

Moreover, according to (6.3.2) and (6.3.5), respectively, we have

(1 − ω)A(1 − ω) = opr(a),

a ∈S(R, Lµ;ℓ

(cl)(X ; R;E,F ))


Hence we conclude

ωAω = ω(ωAω

)ω = ωop

γ−n2

M (h)ω + ωAω,

ωAω ∈ CM+G(,V )(X∧, (γ, (−N, 0]);E,F ),

(1 − ω)A(1 − ω) = (1 − ω)((1 − ω)A(1 − ω)

)(1 − ω) = (1 − ω)opr(a)(1 − ω),

i. e., the tuple (h, a) fulfills (6.3.8).Next let ω1, ω2, ω3 ∈ C∞

0 (R+) be cut-off functions that satisfy (6.3.9). Wemay write

A = ω1Aω2 + (1 − ω1)A(1 − ω3) +(ω1A(1 − ω2) + (1 − ω1)Aω3

),

where(ω1A(1 − ω2) + (1 − ω1)Aω3

)∈ CG(,V )(X

∧, (γ, (−N, 0]);E,F )

according to (6.3.3) and (6.3.6), respectively. Consequently, A is of the form(6.3.10). This finishes the proof of the theorem.

Notation 6.3.8. We refer to any system of cut-off functions ω1, ω2, ω3 satisfying(6.3.9) as subordinated to the covering [0, T1), (T2,∞) of R+.


Definition 6.3.9. Let A ∈ Cµ,;ℓ(V (cl))(X∧, (γ, (−N, 0]);E,F ). We associate with A

the following triple of symbols:

• Complete interior symbol:

Let 0 < T2 < T1 <∞. Any tuple

(h, a) ∈C∞B (R+,M

µ;ℓO(cl)(X ;E,F ))×S(R, Lµ;ℓ

(cl)(X ; R;E,F ))

C∞B (R+,M

µ;ℓV,O(cl)(X ;E,F ))×S(R, Lµ;ℓ

V (cl)(X ; H;E,F ))

that satisfies (6.3.8) in Theorem 6.3.7 is called a complete interior symbolof the operator A, subordinated to the covering [0, T1), (T2,∞) of R+.For short, we write

σµ,;ℓψ,c (A) := (h, a).

Note that the relation A 7→ σµ,;ℓψ,c (A) is non-canonical.• Conormal symbol:

According to Theorem 6.3.4 and Theorem 6.2.21 we associate with Athe tuple

σM (A) = (σ0M (A), . . . , σ

−(N−1)M (A)) ∈ Σµ;ℓ

M(,V (cl))(X, (γ, (−N, 0]);E,F )

of conormal symbols.• Exit symbol:

Let T > 0. According to Theorem 6.3.7 there exists a symbol

a ∈S(R, Lµ;ℓ

(cl)(X ; R;E,F ))


such that

(1 − ω)A(1 − ω) − (1 − ω)opr(a)(1 − ω) ∈ CG(,V )(X∧, (γ, (−N, 0]);E,F )

for all cut-off functions ω, ω ∈ C∞0 (R+) with χ[0,T ] ≺ ω, ω. Any symbol a

that satisfies these conditions is called an exit symbol of the operator A.The exit symbol is regarded as an operator family

σµ,;ℓe (A)(r, τ) := a(r, τ) : Hs(X,E) −→ Hs−µ(X,F )

for τ ∈ R, respectively τ ∈ H, and r ≫ r0 sufficiently large.Note that the relation A 7→ σµ,;ℓe (A) is non-canonical.

Let A ∈ Cµ,;ℓ(V )cl(X∧, (γ, (−N, 0]);E,F ) be a classical (Volterra) cone operator.

• Homogeneous principal symbol:

The (anisotropic) homogeneous principal symbol of A is well-defined asa function

σµ;ℓψ (A) ∈

C∞(R+, S

(µ;ℓ)((T ∗X×R) \ 0,Hom(π∗E, π∗F )))

C∞(R+, S(µ;ℓ)V ((T ∗X×H) \ 0,Hom(π∗E, π∗F ))),


and it has the following properties:

σµ;ℓψ (A)(r, ξx, r

−1τ) ∈C∞(R+, S

(µ;ℓ)((T ∗X×R) \ 0,Hom(π∗E, π∗F )))

C∞(R+, S(µ;ℓ)V ((T ∗X×H) \ 0,Hom(π∗E, π∗F ))),

(1 − ω)σµ;ℓψ (A) ∈

S(R, S(µ;ℓ)((T ∗X×R) \ 0,Hom(π∗E, π∗F )))

S(R, S(µ;ℓ)V ((T ∗X×H) \ 0,Hom(π∗E, π∗F )))

for every cut-off function ω ∈ C∞0 (R+). More precisely, let

A = ω1opγ−n

2

M (h)ω2 + (1 − ω1)opr(a)(1 − ω3) +AM+G

be any representation of A according to (6.3.7). Then

σµ;ℓψ (A)(r, ξx, τ) = ω1(r)σ

µ;ℓψ (h)(r, ξx,

n+ 1

2− γ − i(rτ))

+ (1 − ω1(r))σµ;ℓψ (a)(r, ξx, τ)

(6.3.11)

for r ∈ R+ and (ξx, τ) ∈(T ∗X×R

)\ 0, respectively (ξx, τ) ∈

(T ∗X×H

)\

0. This relation follows from the results concerning the (inverse) Mellinquantization in Theorem 5.2.10 and Theorem 2.6.18; note in particularthe asymptotic expansion formula (2.6.9) in Theorem 2.6.16.

Compositions and adjoints.

Theorem 6.3.10. Let H be another vector bundle over X, and let

A ∈ Cµ,;ℓ(V (cl))(X∧, (γ, (−N, 0]);F,H),

B ∈ Cµ′,′;ℓ

(V (cl))(X∧, (γ, (−N, 0]);E,F ).

Then we haveAB ∈ Cµ+µ′,+′;ℓ

(V (cl)) (X∧, (γ, (−N, 0]);E,H)

for the composition as operators on Sγ(X∧, E).

Let 0 < T2 < T1 < ∞, and let σµ,;ℓψ,c (A) = (h1, a1) and σµ′,′;ℓψ,c (B) =

(h2, a2) be complete interior symbols of A and B subordinated to the covering[0, T1), (T2,∞) of R+. Then

σµ+µ′,+′;ℓψ,c (AB) = (h1#h2, a1#a2)

is a complete interior symbol of the composition AB, subordinated to the covering[0, T1), (T2,∞) of R+. The involved Leibniz-products are according to Theorem5.3.3 and Theorem 5.4.4, respectively.

The following relations hold for the exit symbol and the conormal symbols:

σµ+µ′,+′;ℓe (AB) = σµ,;ℓe (A)#σµ

′,′;ℓe (B),

σ−kM (AB) =

∑

p+q=k

T−qσ−pM (A)σ−q

M (B)

for k = 0, . . . , N − 1. The Leibniz-product of the exit symbols is according toTheorem 5.4.4.


If A and B are classical (Volterra) cone operators, then the (anisotropic)homogeneous principal symbol of the composition is given as

σµ+µ′;ℓψ (AB) = σµ;ℓ

ψ (A)σµ′ ;ℓψ (B).

Proof. Let ω, ω ∈ C∞0 (R+) be cut-off functions near r = 0.

Step 1: Consider the operator ωABω:Choose cut-off functions ω, ω ∈ C∞

0 (R+) such that ω, ω ≺ ω ≺ ω. We maywrite

ωABω =(ωAω

)(ωBω

)+ ωA(1 − ω)Bω.

Now ωA(1− ω) ∈ CG(,V )(X∧, (γ, (−N, 0]);F,H) according to Definition 6.3.2, and

from Corollary 6.3.6 we conclude

ωABω ≡(ωAω

)(ωBω

)mod CG(,V )(X

∧, (γ, (−N, 0]);E,H).

We have

ωAω ≡ ωopγ−n

2

M (h1)ω mod CM+G(,V )(X∧, (γ, (−N, 0]);F,H),

h1 ∈ C∞B (R+,M

µ;ℓ(V,)O(cl)(X ;F,H)),

ωBω ≡ ωopγ−n

2

M (h2)ω mod CM+G(,V )(X∧, (γ, (−N, 0]);E,F ),

h2 ∈ C∞B (R+,M

µ′;ℓ(V,)O(cl)(X ;E,F )).

(1)

To see this choose a cut-off function ω′ ∈ C∞0 (R+) such that ω ≺ ω′. From (6.3.1),

(6.3.4) we obtain

ω′Aω′ = opγ−n

2

M (h1) + A,

A ∈ CM+G(,V )(X∧, (γ, (−N, 0]);F,H),

ω′Bω′ = opγ−n

2

M (h2) + B,

B ∈ CM+G(,V )(X∧, (γ, (−N, 0]);E,F ),

and hence multiplying from the left and from the right with the involved cut-offfunctions ω, ω and ω, ω, respectively, yields (1). Theorem 6.3.3 and Corollary 6.3.6imply

(ωAω

)(ωBω

)≡ ωop

γ−n2

M (h1)ωopγ−n

2

M (h2)ω

modulo CM+G(,V )(X∧, (γ, (−N, 0]);E,H). From Theorem 5.3.3 and Proposition

5.3.4 we conclude

ωopγ−n

2

M (h1)ωopγ−n

2

M (h2)ω=ωopγ−n

2

M (h1#h2)ω−ωopγ−n

2

M (h1)(1 − ω)opγ−n

2

M (h2)ω,

ωopγ−n

2

M (h1)(1 − ω)opγ−n

2

M (h2)ω = ωopγ−n

2

M (h)ω,

h ∈ C∞B (R+,M

−∞(V,)O(X ;E,H)).


Carrying out a Taylor expansion we may write

ωopγ−n

2

M (h)ω =

N−1∑

j=0

ωrjopγ−n

2

M (1

j!(∂jr h)(0, z))ω + ωrNop

γ−n2

M (hN )ω,

1

j!(∂jr h)(0, z) ∈M−∞

(V,)O(X ;E,H), j = 0, . . . , N − 1,

hN ∈ C∞B (R+,M

−∞(V,)O(X ;E,H)),

and from Theorem 5.3.6, Theorem 5.3.3, Proposition 2.6.4 and Proposition 5.3.5(in case of Volterra operators) we obtain

ωrNopγ−n

2

M (hN )ω ∈ CG(,V )(X∧, (γ, (−N, 0]);E,H), i. e.,

ωopγ−n

2

M (h)ω ∈ CM+G(,V )(X∧, (γ, (−N, 0]);E,H).

Summing up, we have shown

ωABω ≡ ωopγ−n

2

M (h1#h2)ω

modulo CM+G(,V )(X∧, (γ, (−N, 0]);E,H), and by Theorem 5.3.2 ωABω is of the

form (6.3.1) or (6.3.4), respectively.If 0 < T2 < T1 < ∞ and ω, ω ∈ C∞

0 (R+) are cut-off functions with χ[0,T2] ≺ω, ω ≺ χ[0,T1] we choose the cut-off functions ω, ω at the beginning of the proofof Step 1 with ω, ω ≺ ω ≺ ω ≺ χ[0,T1]. Hence we see that in (1) we may choosethe holomorphic Mellin symbols h1, h2 as the Mellin components of the completeinterior symbols of A and B subordinated to the covering [0, T1), (T2,∞) ofR+. Consequently, the Leibniz-product h1#h2 serves as the Mellin component ofa complete interior symbol of the composition AB, subordinated to the covering[0, T1), (T2,∞) of R+.Step 2: Consider (1 − ω)AB(1 − ω):

Let ω, ω ∈ C∞0 (R+) be cut-off functions with ω ≺ ω ≺ ω, ω. We may write

(1 − ω)AB(1 − ω) =((1 − ω)A(1 − ω)

)((1 − ω)B(1 − ω)

)+ (1 − ω)AωB(1 − ω),

where (1 − ω)Aω ∈ CG(,V )(X∧, (γ, (−N, 0]);F,H) by Definition 6.3.2, and conse-

quently

(1 − ω)AB(1 − ω) ≡((1 − ω)A(1 − ω)

)((1 − ω)B(1 − ω)

)

mod CG(,V )(X∧, (γ, (−N, 0]);E,H)


due to Corollary 6.3.6. We have

(1 − ω)A(1 − ω) ≡ (1 − ω)opr(a1)(1 − ω)

mod CG(,V )(X∧, (γ, (−N, 0]);F,H),

a1 ∈S(R, Lµ;ℓ

(cl)(X ; R;F,H)

S(R;Lµ;ℓV (cl)(X ; H;F,H),

(1 − ω)B(1 − ω) ≡ (1 − ω)opr(a2)(1 − ω)

mod CG(,V )(X∧, (γ, (−N, 0]);E,F ),

a2 ∈S

′

(R, Lµ′;ℓ

(cl) (X ; R;E,F )

S′

(R;Lµ′;ℓV (cl)(X ; H;E,F ).

(2)

Consequently, we obtain from Theorem 6.3.3, Corollary 6.3.6 and Theorem 5.4.4

(1 − ω)AB(1 − ω) ≡ (1 − ω)opr(a1#a2)(1 − ω) + (1 − ω)opr(a1)ωopr(a2)(1 − ω)

modulo CG(,V )(X∧, (γ, (−N, 0]);E,H), and by Theorem 5.4.3 and Proposition

5.4.5

(1 − ω)opr(a1)ωopr(a2)(1 − ω) = (1 − ω)opr(c)(1 − ω),

c ∈S−∞(R, L−∞(X ; R;E,H)

S−∞(R;L−∞V (X ; H;E,H).

Theorem 5.4.7, Theorem 5.4.4 and Proposition 5.4.6 (in case of Volterra operators)imply

(1 − ω)opr(c)(1 − ω) ∈ CG(,V )(X∧, (γ, (−∞, 0]);E,H).

Summing up, we have shown

(1 − ω)AB(1 − ω) ≡ (1 − ω)opr(a1#a2)(1 − ω)

modulo CG(,V )(X∧, (γ, (−N, 0]);E,H), and by Proposition 6.1.5 and Theorem

5.4.3 (1 − ω)AB(1 − ω) is of the form (6.3.2) or (6.3.5), respectively.Let 0 < T2 < T1 < ∞, and let ω, ω ∈ C∞

0 (R+) be cut-off functions withχ[0,T2] ≺ ω, ω ≺ χ[0,T1]. At the beginning of the proof of Step 2 choose suitablecut-off functions ω, ω such that χ[0,T2] ≺ ω ≺ ω ≺ ω, ω. Hence in (2) we may choosethe symbols a1, a2 as the Fourier components of the complete interior symbols ofA and B, subordinated to the covering [0, T1), (T2,∞) of R+. Consequently,the Leibniz-product a1#a2 is a possible choice of the Fourier component of acomplete interior symbol of the composition AB, subordinated to the covering[0, T1), (T2,∞) of R+. Moreover, we see from the proof that the Leibniz-productof the exit symbols of A and B is an exit symbol of the composition AB.Step 3: Assume ω ≺ ω, and consider the operators ωAB(1− ω) and (1− ω)ABω:


Let ω ∈ C∞0 (R+) be a cut-off function near r = 0 such that ω ≺ ω ≺ ω. We

may write

ωAB(1 − ω) =(ωA(1 − ω)

)B(1 − ω) + ωA

(ωB(1 − ω)

),

(1 − ω)ABω =((1 − ω)Aω

)Bω + (1 − ω)A

((1 − ω)Bω

).

Due to (6.3.3) and (6.3.6), respectively, we have

ωA(1 − ω), (1 − ω)Aω ∈ CG(,V )(X∧, (γ, (−N, 0]);F,H),

ωB(1 − ω), (1 − ω)Bω ∈ CG(,V )(X∧, (γ, (−N, 0]);E,F ),

and from Corollary 6.3.6 we obtain

ωAB(1 − ω), (1 − ω)ABω ∈ CG(,V )(X∧, (γ, (−N, 0]);E,H).

Conclusion: From Step 1 – 3 we see that the composition AB belongs to

Cµ+µ′,+′;ℓ(V (cl)) (X∧, (γ, (−N, 0]);E,H) in view of Definition 6.3.2. Moreover, the for-

mula for the complete interior symbol of AB subordinated to a given covering[0, T1), (T2,∞) of R+, as well as the relationship for the exit symbol, are provedin Step 1 and Step 2. The identities for the conormal symbols of AB are subjectto Theorem 6.2.21, see also Theorem 6.3.4.

For classical operators the homogeneous principal symbol of the compositionis given as the product of the homogeneous principal symbols. This follows fromthe relationship for the complete interior symbol of AB and equation (6.3.11),keeping in mind the asymptotic expansion formulae for the Leibniz-products.

Theorem 6.3.11. Let A ∈ Cµ,;ℓ(cl) (X∧, (γ, (−N, 0]);E,F ). Then the formal adjoint

A∗ with respect to the r−n2 L2-inner product is a cone pseudodifferential operator

in Cµ,;ℓ(cl) (X∧, (−γ, (−N, 0]);F,E).

Let 0 < T2 < T1 < ∞, and let σµ,;ℓψ,c (A) = (h, a) be a complete interior

symbol of A subordinated to the covering [0, T1), (T2,∞) of R+. Then

σµ,;ℓψ,c (A∗) = (h∗, a(∗),n2 )

is a complete interior symbol of A∗ subordinated to the covering [0, T1), (T2,∞)of R+. The adjoint symbols are according to Theorem 5.3.3 and Theorem 5.4.4,respectively.

The following relations hold for the exit symbol and the conormal symbols:

σµ,;ℓe (A∗) = σµ,;ℓe (A)(∗),n2 ,

σ−kM (A∗)(z) = (σ−k

M (A)(n+ 1 − k − z))(∗)

for k = 0, . . . , N − 1, where (∗) denotes the formal adjoint with respect to the L2-inner product on the manifold X in the formula for the conormal symbols, and theadjoint exit symbol is according to Theorem 5.4.4.

If A is a classical cone operator, then the (anisotropic) homogeneous principalsymbol of A∗ is given as

σµ;ℓψ (A∗) = σµ;ℓ

ψ (A)∗.


Proof. Let ω, ω ∈ C∞0 (R+) be cut-off functions near r = 0. According to (6.3.1)

we may write

ωAω = ωopγ−n

2

M (h)ω +AM+G,

AM+G ∈ CM+G(X∧, (γ, (−N, 0]);E,F ),

h ∈ C∞B (R+,M

µ;ℓO(cl)(X ;E,F )).

HenceωA∗ω = ωop

−γ−n2

M (h∗)ω +A∗M+G

with the adjoint symbol h∗ from Theorem 5.3.3. Note that due to Theorem 6.2.21we have A∗

M+G ∈ CM+G(X∧, (−γ, (−N, 0]);F,E). Consequently, ωA∗ω is of theform (6.3.1) by Theorem 5.3.2.

In view of (6.3.2) we have

(1 − ω)A(1 − ω) = (1 − ω)opr(a)(1 − ω) +G,

G ∈ CG(X∧, (γ, (−N, 0]);E,F ),

a ∈ S(R, Lµ;ℓ(cl)(X ; R;E,F )),

and thus(1 − ω)A∗(1 − ω) = (1 − ω)opr(a

(∗),n2 )(1 − ω) +G∗,

where G∗ ∈ CG(X∧, (−γ, (−N, 0]);F,E), and a(∗),n2 is the adjoint symbol to afrom Theorem 5.4.4. Proposition 6.1.5 and Theorem 5.4.3 imply that (1−ω)A∗(1−ω) is of the form (6.3.2).

Next assume that ω ≺ ω. Then also ω ≺ ω, and thus

(1 − ω)Aω, ωA(1 − ω) ∈ CG(X∧, (γ, (−N, 0]);E,F )

by (6.3.3). Consequently,

ωA∗(1 − ω), (1 − ω)A∗ω ∈ CG(X∧, (−γ, (−N, 0]);F,E).

From Definition 6.3.2 we obtain that A∗ ∈ Cµ,;ℓ(cl) (X∧, (−γ, (−N, 0]);F,E) as de-

sired. Moreover, the formulae for the complete interior symbol of A∗ subordinatedto a covering [0, T1), (T2,∞) of R+, as well as for the exit symbol, follow im-

mediately from the proof. In the classical case, the identity σµ;ℓψ (A∗) = σµ;ℓ

ψ (A)∗

for the homogeneous principal symbol is a consequence of (6.3.11), the relation forthe complete interior symbol, and the asymptotic expansions of h∗ in terms of h,and of a(∗),n2 in terms of a, from Theorem 5.3.3 and Theorem 5.4.4, respectively.The assertion concerning the conormal symbols follows from Theorem 6.3.4 andTheorem 6.2.21.

6.4. Ellipticity and Parabolicity

Definition 6.4.1. Let A ∈ Cµ,;ℓ(X∧, (γ, (−N, 0]);E,F ).

a) Interior ellipticity:

A is called elliptic in the interior, respectively interior elliptic, if for eachT > 0 the following conditions are fulfilled:


• For all cut-off functions ω, ω ∈ C∞0 (R+) such that χ[0,T ] ≺ ω, ω, and all

representations

ωAω = opγ−n

2

M (h) +AM+G,

AM+G ∈ CM+G(X∧, (γ, (−N, 0]);E,F ),

h ∈ C∞B (R+,M

µ;ℓO (X ;E,F ))

according to (6.3.1), we require that h is elliptic on the interval [0, T ] inthe sense of Definition 5.3.8.

• For all cut-off functions ω, ω ∈ C∞0 (R+) such that ω, ω ≺ χ[0,T ] write

(1 − ω)A(1 − ω) = opr(a),

a ∈ S(R, Lµ;ℓ(X ; R;E,F ))

according to (6.3.2). We require that a is interior elliptic on the interval[T,∞) in the sense of Definition 5.4.9.

b) Conormal ellipticity:

A is called conormal elliptic if there exists s0 ∈ R such that the conormalsymbol

σ0M (A)(z) : Hs0(X,E) −→ Hs0−µ(X,F )

is an isomorphism for all z ∈ Γn+12 −γ .

c) Exit ellipticity:

A is called exit elliptic if the following is fulfilled: For all cut-off functionsω, ω ∈ C∞


(1 − ω)A(1 − ω) = opr(a),

a ∈ S(R, Lµ;ℓ(X ; R;E,F ))

according to (6.3.2). We require that there exists s0 ∈ R and r0 ∈ R+, suchthat for r > r0 and all τ ∈ R

a(r, τ) : Hs0(X,E) −→ Hs0−µ(X,F )

is an isomorphism, and we have for some M ∈ R

‖a(r, τ)−1‖L(Hs0−µ,Hs0 )〈τ〉M 〈r〉 = O(1),

uniformly for τ ∈ R and r → ∞.

A is called elliptic if A is interior elliptic, conormal elliptic, and exit elliptic.

Definition 6.4.2. Let A ∈ Cµ,;ℓV (X∧, (γ, (−N, 0]);E,F ).

a) Interior parabolicity:

A is called parabolic in the interior, respectively interior parabolic, if for eachT > 0 the following is fulfilled:


• For all cut-off functions ω, ω ∈ C∞0 (R+) such that χ[0,T ] ≺ ω, ω, and all

representations

ωAω = opγ−n

2

M (h) +AM+G,

AM+G ∈ CM+G,V (X∧, (γ, (−N, 0]);E,F ),

h ∈ C∞B (R+,M

µ;ℓV,O(X ;E,F ))

according to (6.3.4), we require that h is parabolic on the interval [0, T ] inthe sense of Definition 5.3.8.

• For all cut-off functions ω, ω ∈ C∞0 (R+) such that ω, ω ≺ χ[0,T ] write

(1 − ω)A(1 − ω) = opr(a),

a ∈ S(R, Lµ;ℓV (X ; H;E,F ))

according to (6.3.5). We require that a is interior parabolic on the interval[T,∞) in the sense of Definition 5.4.9.

b) Conormal parabolicity:

A is called conormal parabolic if there exists s0 ∈ R such that the conormalsymbol

σ0M (A)(z) : Hs0(X,E) −→ Hs0−µ(X,F )

is an isomorphism for all z ∈ Hn+12 −γ .

c) Exit parabolicity:

A is called exit parabolic if the following is fulfilled: For all cut-off functionsω, ω ∈ C∞


(1 − ω)A(1 − ω) = opr(a),

a ∈ S(R, Lµ;ℓV (X ; H;E,F ))

according to (6.3.5). We require that there exists s0 ∈ R and r0 ∈ R+, suchthat for r > r0 and all z ∈ H

a(r, z) : Hs0(X,E) −→ Hs0−µ(X,F )


‖a(r, z)−1‖L(Hs0−µ,Hs0)〈z〉M 〈r〉 = O(1),

uniformly for z ∈ H and r → ∞.

A is called parabolic if A is interior parabolic, conormal parabolic, and exit para-bolic.

Notation 6.4.3. For ∈ R we denote

C−∞,(V ) (X∧, (γ, (−N, 0]);E,F ) :=

⋂

µ∈R

Cµ,;ℓ(V ) (X∧, (γ, (−N, 0]);E,F ).

Consequently, A ∈ C−∞,(V ) (X∧, (γ, (−N, 0]);E,F ) if and only if the following holds:


• For all cut-off functions ω, ω ∈ C∞0 (R+) we have

ωAω ∈ CM+G(,V )(X∧, (γ, (−N, 0]);E,F ),

(1 − ω)A(1 − ω) = opr(a),

a ∈S(R, L−∞(X ; R;E,F )

S(R, L−∞V (X ; H;E,F ).

• For all cut-off functions ω ≺ ω

ωA(1 − ω), (1 − ω)Aω ∈ CG(,V )(X∧, (γ, (−N, 0]);E,F ).

Moreover, let C−∞,(V ) (X∧, (γ, (−N, 0]);E,F )0 denote the subspace of all those op-

erators A with σ−kM (A) = 0 for k = 0, . . . , N − 1.

Theorem 6.4.4. Let A ∈ Cµ,;ℓ(V (cl))(X∧, (γ, (−N, 0]);E,F ), and let 0 < T2 < T1 <

∞. Moreover, let σµ,;ℓψ,c (A) = (h, a) be a complete interior symbol of A, subordi-

nated to the covering [0, T1), (T2,∞) of R+.The following are equivalent:

a) A is interior elliptic (parabolic).

b) There exist T2 < T2 < T1 < T1 < ∞ such that h is elliptic (parabolic) on

the interval [0, T1] in the sense of Definition 5.3.8, and a is interior elliptic

(parabolic) on the interval [T2,∞) in the sense of Definition 5.4.9.

c) There exists P ∈ C−µ,−;ℓ(V (cl)) (X∧, (γ, (−N, 0]);F,E) such that

AP − 1 ∈ C−∞,0(V ) (X∧, (γ, (−N, 0]);F ),

PA− 1 ∈ C−∞,0(V ) (X∧, (γ, (−N, 0]);E).

Moreover, if A ∈ Cµ,;ℓ(V ) cl(X∧, (γ, (−N, 0]);E,F ), then A is interior elliptic (para-

bolic) if and only if the homogeneous principal symbol σµ;ℓψ (A) satisfies the follow-

ing:

σµ;ℓψ (A)(r, ξx, r

−1τ) is invertible in Hom(π∗E, π∗F ) for all r ∈ R+ and 0 6=(ξx, τ) ∈ T ∗X×R (respectively 0 6= (ξx, τ) ∈ T ∗X×H), and for the inverse we have

‖σµ;ℓψ (A)(r, ξx, τ)

−1‖〈r〉 = O(1),

uniformly for(|ξx|2ℓx + |τ |2

)= 1 and r → ∞.

Proof. c) ⇒ a): Let (h1, a1) and (h2, a2) be any choices of complete interior sym-bols of the operators A and P , respectively, subordinated to the same covering[0, T ′

1), (T′2,∞) of R+. Then for every choice of cut-off functions ω ≺ ω ≺ χ[0,T ′

1]

we have

ω(op

γ−n2

M (h1)opγ−n

2

M (h2) − 1)ω ∈ CM+G(,V )(X

∧, (γ, (−N, 0]);F ),

ω(op

γ−n2

M (h2)opγ−n

2

M (h1) − 1)ω ∈ CM+G(,V )(X

∧, (γ, (−N, 0]);E).


From Lemma 6.2.15, Proposition 6.1.5 and Theorem 5.3.11 (respectively Theorem5.3.10) we conclude that h1 is elliptic (parabolic) on every interval (0, T ] such that

T < T ′1, where h1 is regarded as an element of C∞

B (R+, Lµ;ℓ(cl)(X ; Γn+1

2 −γ ;E,F )),

respectively C∞B (R+, L

µ;ℓV (cl)(X ; Hn+1

2 −γ ;E,F )). Moreover, Theorem 6.3.10 im-

plies that the conormal symbol σ0M (A) is elliptic (parabolic) as a meromorphic

(Volterra) Mellin symbol in the sense of Definition 5.1.12 or Definition 5.2.7, re-spectively. Since h1(0, z) ≡ σ0

M (A) modulo meromorphic (Volterra) Mellin symbolsof order −∞ we finally obtain that h1 is elliptic (parabolic) on the interval [0, T ]

as an element of C∞B (R+,M

µ;ℓ(V,)O(cl)(X ;E,F )). Similarly, we conclude that a1 is

interior elliptic (parabolic) on the interval [T,∞), for every T ′2 < T . Hence, us-

ing Theorem 5.3.2 and Theorem 5.4.3, the interior ellipticity (parabolicity) of theoperator A follows.

a) ⇒ b): Choose arbitrary T2 < T2 < T1 < T1 < ∞. Let ω, ω ∈ C∞0 (R+)

such that χ[0,T1]≺ ω ≺ ω ≺ χ[0,T1]. We may write

ωAω = ωopγ−n

2

M (h)ω +AM+G,

AM+G ∈ CM+G(,V )(X∧, (γ, (−N, 0]);E,F ).

Let h′ ∈ C∞B (R+,M

µ;ℓ(V,)O(cl)(X ;E,F )) be the left-symbol that is associated with

the double-symbol ω(r)h(r, z)ω(r′) according to Theorem 5.3.2. Then we have

ωAω = opγ−n

2

M (h′) + AM+G, and by Definition 6.4.1, respectively Definition

6.4.2, the symbol h′ is elliptic (parabolic) on the interval [0, T1]. For h′ − ωh ∈C∞B (R+,M

µ−1;ℓ(V,)O(cl)(X ;E,F )), we conclude that h is elliptic (parabolic) on the in-

terval [0, T1]. Next let ω, ω ∈ C∞0 (R+) such that χ[0,T2] ≺ ω ≺ ω ≺ χ[0,T2]

. Then

(1 − ω)A(1 − ω) = (1 − ω)opr(a)(1 − ω) +G,

G ∈ CG(,V )(X∧, (γ, (−N, 0]);E,F ).

Let a′ be the left-symbol associated with the double-symbol (1 − ω(r))a(r, τ)(1 −ω(r′)), see Theorem 5.4.3. Due to Proposition 6.1.5 write G = opr(g), and thus

(1 − ω)A(1 − ω) = opr(a′ + g),

a′ + g ∈S(R, Lµ;ℓ

(cl)(X ; R;E,F )),


By Definition 6.4.1, respectively Definition 6.4.2, a′ + g is interior elliptic (para-

bolic) on the interval [T2,∞). For

(a′ + g) − (1 − ω)a ∈S−1(R, Lµ−1;ℓ

(cl) (X ; R;E,F )),

S−1(R, Lµ−1;ℓV (cl) (X ; H;E,F )),

we obtain that a is interior elliptic (parabolic) on [T2,∞).


b) ⇒ c): Due to Theorem 5.3.11 (respectively Theorem 5.3.10) and Theorem5.4.11 there exist

h ∈ C∞B (R+,M

−µ;ℓ(V,)O(cl)(X ;F,E)),

a ∈S−(R, L−µ;ℓ

(cl) (X ; R;F,E)),

S−(R, L−µ;ℓV (cl)(X ; H;F,E)),

such that for all cut-off functions ω, ω ≺ χ[0,T1]and χ[0,T2]

≺ ω, ω we have

ω(opγ−n

2

M (h)opγ−n

2

M (h) − 1)ω ∈ CM+G(,V )(X

∧, (γ, (−N, 0]);F ),

ω(opγ−n

2

M (h)opγ−n

2

M (h) − 1)ω ∈ CM+G(,V )(X

∧, (γ, (−N, 0]);E),

(1 − ω)(opr(a)opr(a) − 1

)(1 − ω) = (1 − ω)opr(rR)(1 − ω),

rR ∈S0(R, L−∞(X ; R;F, F )),

S0(R, L−∞V (X ; H;F, F )),

(1 − ω)(opr(a)opr(a) − 1

)(1 − ω) = (1 − ω)opr(rL)(1 − ω),

rL ∈S0(R, L−∞(X ; R;E,E)),

S0(R, L−∞V (X ; H;E,E)).

(1)

Notice that for all ϕ, ψ ∈ C∞0 (T2, T1) we have ϕop

γ−n2

M (h)ψ ≡ ϕopr(a)ψ mod-

ulo CG(,V )(X∧, (γ, (−N, 0]);E,F ). Now we conclude that also ϕop

γ−n2

M (h)ψ ≡ϕopr(a)ψ modulo CG(,V )(X

∧, (γ, (−N, 0]);F,E), for all ϕ, ψ ∈ C∞0 (T2, T1). Let

ω1, ω2, ω3 be cut-off functions subordinated to the covering [0, T1), (T2,∞) of

R+, and define P := ω1opγ−n

2

M (h)ω2 + (1 − ω1)opr(a)(1 − ω3).

According to Theorem 6.3.3 we have P ∈ C−µ,−;ℓ(V (cl)) (X∧, (γ, (−N, 0]);F,E).

Moreover, σ−µ,−;ℓψ,c (P ) = (h, a) is a complete interior symbol of P subordinated

to the covering [0, T1), (T2,∞) of R+. Theorem 6.3.10 and (1) imply

ω1(AP − 1)ω2 ≡ ω1

(opγ−n

2

M (h#h) − 1)ω2 ≡ 0

mod CM+G(,V )(X∧, (γ, (−N, 0]);F ),

ω1(PA− 1)ω2 ≡ ω1

(opγ−n

2

M (h#h) − 1)ω2 ≡ 0

mod CM+G(,V )(X∧, (γ, (−N, 0]);E),

(1 − ω1)(AP − 1)(1 − ω3) ≡ (1 − ω1)(opr(a#a) − 1

)(1 − ω3)

≡ (1 − ω1)opr(rR)(1 − ω3)

mod CG(,V )(X∧, (γ, (−N, 0]);F ),

(1 − ω1)(PA− 1)(1 − ω3) ≡ (1 − ω1)(opr(a#a) − 1

)(1 − ω3)

≡ (1 − ω1)opr(rL)(1 − ω3)

mod CG(,V )(X∧, (γ, (−N, 0]);E),

(2)


i. e., P fulfills the conditions in c).Interior ellipticity (parabolicity) for classical operators: According to (6.3.11)

we have

σµ;ℓψ (h)(r, ξx,

n+ 1

2− γ − iτ) = σµ;ℓ

ψ (A)(r, ξx, r−1τ) on [0, T1],

σµ;ℓψ (a)(r, ξx, τ) = σµ;ℓ

ψ (A)(r, ξx, τ) on [T2,∞),

for (ξx, τ) ∈(T ∗X×R

)\ 0, respectively (ξx, τ) ∈

(T ∗X×H

)\ 0. Consequently, the

asserted equivalence for classical (cone) operators follows from b) and Definition5.3.8, as well as Definition 5.4.9.

Theorem 6.4.5. Let A ∈ Cµ,;ℓ(V (cl))(X∧, (γ, (−N, 0]);E,F ). The following are

equivalent:

a) A is interior and exit elliptic (parabolic).b) A is interior elliptic (parabolic), and for some (every) exit symbol σµ,;ℓe (A) we

have:There exists s0 ∈ R such that for r ≫ r0 sufficiently large and all τ ∈ R

(respectively τ ∈ H)

σµ,;ℓe (A)(r, τ) : Hs0(X,E) −→ Hs0−µ(X,F )


‖σµ,;ℓe (A)(r, τ)−1‖L(Hs0−µ,Hs0 )〈τ〉M 〈r〉 = O(1),

uniformly for τ ∈ R (respectively τ ∈ H), and r → ∞.

c) There exists P ∈ C−µ,−;ℓ(V (cl)) (X∧, (γ, (−N, 0]);F,E) such that

AP − 1 ∈ CM+G(,V )(X∧, (γ, (−N, 0]);F ),

PA− 1 ∈ CM+G(,V )(X∧, (γ, (−N, 0]);E).

Moreover, the following are equivalent:

i) A is interior and conormal elliptic (parabolic).

ii) There exists P ∈ C−µ,−;ℓ(V (cl)) (X∧, (γ, (−N, 0]);F,E) such that

AP − 1 ∈ C−∞,0(V ) (X∧, (γ, (−N, 0]);F )0,

PA− 1 ∈ C−∞,0(V ) (X∧, (γ, (−N, 0]);E)0.

Proof. We first prove the equivalence a) – c): Note that if e1 and e2 are exitsymbols for the operator A, then there exist cut-off functions ω ≺ ω such that(1 − ω)e1(1 − ω) ≡ (1 − ω)e2(1 − ω) modulo terms of order −∞, both in µ and. Hence the condition in b) does not depend on the choice of the exit symbol.In particular, we may consider the Fourier component a of any complete interiorsymbol (h, a) as an exit symbol that satisfies b).

Let 0 < T2 < T1 < ∞, and let (h, a) be a complete interior symbol of A,subordinated to the covering [0, T1), (T2,∞) of R+. From Theorem 6.4.4 we

obtain that a) and b) are equivalent to the following: There exist T2 < T2 < T1 <


T1 < ∞ such that h is elliptic (parabolic) on the interval [0, T1] in the sense of

Definition 5.3.8, and a is elliptic (parabolic) on the interval [T2,∞) in the sense ofDefinition 5.4.9.

Starting from this condition we construct the operator P in c) analogouslyto the proof of b) ⇒ c) in Theorem 6.4.4. Note that in (1) we now obtain

rR ∈S−∞(R, L−∞(X ; R;F, F ))

S−∞(R, L−∞V (X ; H;F, F )),

rL ∈S−∞(R, L−∞(X ; R;E,E))

S−∞(R, L−∞V (X ; H;E,E)),

and consequently in (2)

(1 − ω1)(AP − 1)(1 − ω3) ∈ CG(,V )(X∧, (γ, (−N, 0]);F ),

(1 − ω1)(PA− 1)(1 − ω3) ∈ CG(,V )(X∧, (γ, (−N, 0]);E).

This shows that P satisfies the conditions in c).c) ⇒ a) is obvious due to Theorem 6.4.4.Let us now prove the equivalence i) ⇔ ii): ii) ⇒ i) follows from The-

orem 6.4.4 and Theorem 6.3.10. Now assume that i) holds. Choose P ′ ∈C−µ,−;ℓ

(V (cl)) (X∧, (γ, (−N, 0]);F,E) that satisfies c) in Theorem 6.4.4. Hence we obtain

from the conormal ellipticity (parabolicity), using Theorem 5.1.14 and Theorem5.2.8:

g := σ0M (A)−1 − σ0

M (P ′) ∈M−∞Q (X ;F,E), πCQ∩Γn+1

2 −γ = ∅,M−∞V,Q (X ; Hn+1

2 −γ ;F,E).

With a cut-off function ω ∈ C∞0 (R+) define

P := P ′ + ωopγ−n

2

M (g)ω ∈ C−µ,−;ℓ(V (cl)) (X∧, (γ, (−N, 0]);F,E).

Then we have AP = 1 + C1, and PA = 1 + C2, where the remainders

C1 ∈ C−∞,0(V ) (X∧, (γ, (−N, 0]);F ),

C2 ∈ C−∞,0(V ) (X∧, (γ, (−N, 0]);E)

are such that σ0M (C1) = 0, as well as σ0

M (C2) = 0. Consider the operator C1: Wemay write

C1 = C1 + C1,

C1 ∈ CM+G(,V )(X∧, (γ, (−N, 0]);F ),

C1 ∈ C−∞,0(V ) (X∧, (γ, (−N, 0]);F )0.

From Theorem 6.2.27 we conclude that there exists a smoothing (Volterra) Mellin

and Green operator D1 ∈ CM+G(,V )(X∧, (γ, (−N, 0]);F ) such that

(1 + C1)(1 + D1) − 1 ∈ CG(,V )(X∧, (γ, (−N, 0]);F ).

Note that C−∞,0(V ) (X∧, (γ, (−N, 0]);F )0 is a two-sided ideal in the (Volterra) cone

algebra. Hence we obtain APR − 1 ∈ C−∞,0(V ) (X∧, (γ, (−N, 0]);F )0 with PR :=


P (1 + D1). Analogously, we construct PL ∈ C−µ,−;ℓ(V (cl)) (X∧, (γ, (−N, 0]);F,E) with

PLA − 1 ∈ C−∞,0(V ) (X∧, (γ, (−N, 0]);E)0. But both PL and PR differ only by an

element in C−∞,0(V ) (X∧, (γ, (−N, 0]);F,E)0, and thus condition ii) is fulfilled with

either P := PL or P := PR. This finishes the proof of the theorem.

Theorem 6.4.6. Let A ∈ Cµ,;ℓ(cl) (X∧, (γ, (−N, 0]);E,F ). The following are equiv-

alent:

a) A is elliptic.

b) There exists P ∈ C−µ,−;ℓ(cl) (X∧, (γ, (−N, 0]);F,E) such that AP = 1 +G1 and

PA = 1 + G2 with Green operators G1 ∈ CG(X∧, (γ, (−N, 0]);F ) and G2 ∈CG(X∧, (γ, (−N, 0]);E).

Moreover, for A ∈ Cµ,;ℓV (cl)(X∧, (γ, (−N, 0]);E,F ) the following are equivalent:

i) A is parabolic.

ii) There exists P ∈ C−µ,−;ℓV (cl) (X∧, (γ, (−N, 0]);F,E) such that AP = 1+G1 and

PA = 1 +G2 with Volterra Green operators G1 ∈ CG,V (X∧, (γ, (−N, 0]);F )and G2 ∈ CG,V (X∧, (γ, (−N, 0]);E).

iii) There exists P ∈ C−µ,−;ℓV (cl) (X∧, (γ, (−N, 0]);F,E) such that AP = 1 and

PA = 1, i. e., A is invertible in the Volterra cone algebra with A−1 = P .

Proof. We simultaneously prove the equivalences a) ⇔ b) and i) ⇔ ii).Clearly, the conditions in ii) and b) are sufficient for i) and a) by Theorem6.4.4 and Theorem 6.4.5. Now assume that A is elliptic (parabolic). Usingthe interior and exit ellipticity (parabolicity), we conclude from Theorem 6.4.5

that there exists P ′ ∈ C−µ,−;ℓ(V (cl)) (X∧, (γ, (−N, 0]);F,E) such that AP ′ − 1 ∈

CM+G(,V )(X∧, (γ, (−N, 0]);F ) and P ′A− 1 ∈ CM+G(,V )(X

∧, (γ, (−N, 0]);E). Us-ing the conormal ellipticity (parabolicity) of A we see that

g := σ0M (A)−1 − σ0

M (P ′) ∈M−∞Q (X ;F,E), πCQ∩Γn+1

2 −γ = ∅,M−∞V,Q (X ; Hn+1

2 −γ ;F,E).

With a cut-off function ω ∈ C∞0 (R+) define

P := P ′ + ωopγ−n

2

M (g)ω ∈ C−µ,−;ℓ(V (cl)) (X∧, (γ, (−N, 0]);F,E).

Then

AP − 1 ∈ CM+G(,V )(X∧, (γ, (−N, 0]);F ),

PA− 1 ∈ CM+G(,V )(X∧, (γ, (−N, 0]);E).

Moreover, we have σ0M (AP ) = 1, as well as σ0

M (PA) = 1, in view of Theorem6.3.10. From Theorem 6.2.27 we obtain the existence of

D1 ∈ CM+G(,V )(X∧, (γ, (−N, 0]);E), D2 ∈ CM+G(,V )(X

∧, (γ, (−N, 0]);F )


such that((1+D1)P

)A−1 ∈ CG(,V )(X

∧, (γ, (−N, 0]);E) and A(P (1+D2)

)−1 ∈

CG(,V )(X∧, (γ, (−N, 0]);F ), and thus either P := P (1 +D2) or P := (1 + D1)P

fulfills the conditions in ii) and b).It remains to prove that iii) is equivalent to ii), but this follows from Theorem

6.1.6.

Definition 6.4.7. Let A ∈ Cµ,;ℓ(V (cl))(X∧, (γ, (−N, 0]);E,F ) be elliptic, respec-

tively parabolic. Then any operator P ∈ C−µ,−;ℓ(V (cl)) (X∧, (γ, (−N, 0]);F,E) that sat-

isfies b) or ii) in Theorem 6.4.6 is called a (Volterra) parametrix of the operatorA.

Corollary 6.4.8. Let A ∈ Cµ,;ℓ(X∧, (γ, (−N, 0]);E,F ) be elliptic. Then


is a Fredholm operator for all s, t, δ ∈ R.

Proof. This follows from Theorem 6.4.6, and from the fact that Green operatorsinduce nuclear, in particular compact, operators in the cone Sobolev spaces byProposition 6.1.4.

Corollary 6.4.9. a) Let A ∈ Cµ,;ℓV (X∧, (γ, (−N, 0]);E,F ) be parabolic. Then


is bijective for all s, t, δ ∈ R, i. e., the equation Au = f with f ∈K(s−µ,t),γ;ℓ(X∧, F )δ− is uniquely solvable with solution u ∈ K(s,t),γ;ℓ(X∧, E)δ.

Moreover, if f has asymptotics of “length” N , i. e., if there exists asuitable asymptotic type Q ∈ As

((γ, (−N, 0]), C∞(X,F )

)such that f ∈

K(s−µ,t),γ;ℓQ (X∧, F )δ−, then the solution u belongs to the space K(s,t),γ;ℓ

Q(X∧, E)

with some resulting asymptotic type Q ∈ As((γ, (−N, 0]), C∞(X,E)

).

b) If A ∈ Cµ,;ℓV (cl)(X∧, (γ, (−N, 0]);E,F ) is just interior and conormal parabolic we

still have the following:For every r0 ∈ R+ the operator


0 ((0, r0]×X,F )

is bijective for all s, t ∈ R.

More precisely, there exists P ∈ C−µ,−;ℓV (cl) (X∧, (γ, (−N, 0]);F,E) such that

(A|

H(s,t),γ;ℓ0 ((0,r0]×X,E)

)−1

= P |H

(s−µ,t),γ;ℓ0 ((0,r0]×X,F )

for all s, t ∈ R.

In particular, the equation Au = f with f ∈ H(s−µ,t),γ;ℓ0 ((0, r0]×X,F ) is

uniquely solvable with solution u ∈ H(s,t),γ;ℓ0 ((0, r0]×X,E). Moreover, if f has

asymptotics of “length” N , then so does u (in the sense of a)).


Proof. a) follows immediately from Theorem 6.4.6. Let us prove b). Choose

P ′ ∈ C−µ,−;ℓV (cl) (X∧, (γ, (−N, 0]);F,E) satisfying condition ii) in Theorem 6.4.5.

Moreover, let ω, ω ∈ C∞0 (R+) be cut-off functions such that χ[0,r0] ≺ ω ≺ ω. Then

we have ωAP ′ω = ω + G1 and ωP ′Aω = ω + G2 with Volterra Green operatorsG1 ∈ CG,V (X∧, (γ, (−N, 0]);F ) and G2 ∈ CG,V (X∧, (γ, (−N, 0]);E). By Theorem

6.1.6 we have (1 +G1)−1 = 1 + G1 and (1 +G2)

−1 = 1 + G2 with Volterra Green

operators G1, G2. Hence we see

ωA(P ′ω(1 + G1)

)= 1 − (1 − ω)(1 + G1),

((1 + G2)ωP

′)Aω = 1 − (1 + G2)(1 − ω).

Restricting to the H(s,t),γ;ℓ0 ((0, r0]×X, ·)-spaces reveals that the so-obtained op-

erator A|H

(s,t),γ;ℓ0 ((0,r0]×X,E)

is indeed invertible from the left and from the right

with operators in C−µ,−;ℓV (cl) (X∧, (γ, (−N, 0]);F,E), and consequently any left- or

right-inverse gives rise to an operator P which satisfies the assertion in b).

Parabolic reductions of orders.

Theorem 6.4.10. Let ℓ ∈ 2N. For s, δ ∈ R there exist

Rs,δ ∈ Cs,δ;ℓV cl (X∧, (γ, (−N, 0]);E)

such that Rs,δR−s,−δ = 1, i. e., there exist parabolic reductions of orders withinthe algebra of classical Volterra cone operators.

Proof. For s = δ = 0 define R0,0 := 1. Now assume that s > 0. Since ℓ ∈ 2N iseven, the function

((ξx, ζ) 7→

(|ξx|ℓx + ζ

) sℓ · idπ∗E(ξx,ζ)

)∈ S

(s;ℓ)V ((T ∗X×H0) \ 0,Hom(π∗E))

is well-defined, and(|ξx|ℓx + ζ

) sℓ 6= 0 for (ξx, ζ) ∈ (T ∗X×H0) \ 0.

According to Theorem 3.2.16 there exists

h0 ∈ Ls;ℓV cl(X ; H0;E),

σs;ℓψ (h0)(ξx, ζ) =(|ξx|ℓx + ζ

) sℓ · idπ∗E(ξx,ζ)

.

Let ψ ∈ C∞0 (R+) such that ψ ≡ 1 near r = 1. With the Mellin kernel cut-off

operator define h1 := H 12(ψ)h0 ∈ M s;ℓ

V,O cl(X ;E). Due to Theorem 5.2.5 we have

h1−h0 ∈ L−∞V (X ; H0;E), and thus h1 is parabolic as an element of M s;ℓ

V,O cl(X ;E)in the sense of Definition 5.2.7. Using Theorem 5.2.8 we obtain that for some β ∈ R

the holomorphic Volterra Mellin symbol h := Tβh1 ∈ M s;ℓV,O cl(X ;E) is parabolic,

and additionally h(z) : Hs0(X,E) −→ Hs0−s(X,E) is an isomorphism for allz ∈ Hn+1

2 −γ and all s0 ∈ R.

From Theorem 3.2.16 and Theorem 3.2.19 we see that there exists

b ∈ Ls;ℓV cl(X ; H;E),

σs;ℓψ (b)(ξx, τ) =(|ξx|ℓx − iτ

) sℓ · idπ∗E(ξx,τ)

,


and b is invertible with inverse b−1 ∈ L−s;ℓV cl (X ; H;E).

Let ω3 ≺ ω1 ≺ ω2 be cut-off functions near r = 0. Moreover, let ϕ ∈ C∞(R+)be an everywhere positive function with ω2 ≺ ϕ, and rϕ(r) ≡ 1 for r ≫ r0sufficiently large. Define

Rs,0 := ω1opγ−n

2

M (h)ω2 + (1 − ω1)opr(a)(1 − ω3),

where a(r, τ) := (1 − ω3(r))b(rϕ(r)τ) ∈ S0(R, Ls;ℓV cl(X ; H;E)). By Theorem 6.3.3

we have Rs,0 ∈ Cs,0;ℓV cl (X∧, (γ, (−N, 0]);E), and from the construction we see

σs;ℓψ (Rs,0)(r, ξx, τ) =(|ξx|ℓx − i(rϕ(r)τ)

) sℓ · idπ∗E(ξx,τ)

,

σ0M (Rs,0) = h,

σs,0;ℓe (Rs,0)(r, τ) = b(τ).

Hence Rs,0 is parabolic, and by Theorem 6.4.6 there exists

R−s,0 :=(Rs,0

)−1 ∈ C−s,0;ℓV cl (X∧, (γ, (−N, 0]);E).

This completes the proof in the case δ = 0.With a cut-off function ω ∈ C∞

0 (R+) we define for all s ∈ R and δ > 0

Rs,δ :=(ω(r) + (1 − ω(r))〈r〉δ

)Rs,0 ∈ Cs,δ;ℓV cl (X

∧, (γ, (−N, 0]);E).

Consequently, Rs,δ is parabolic, and by Theorem 6.4.6 there exists

R−s,−δ :=(Rs,δ

)−1 ∈ C−s,−δ;ℓV cl (X∧, (γ, (−N, 0]);E).


Chapter 7. Remarks on the classical theory of parabolic PDE

We want to conclude the present exposition with some remarks about the clas-sical theory of parabolic partial differential equations; more precisely, we wantto give an idea of how it fits into the framework of our Volterra cone calculus.In particular, the intention of this chapter is to offer the reader some guide tothe functional analytic structures of the previous chapters. To this end, we shalldiscuss parabolicity, solvability, and regularity for a generalized heat operator.

A generalization of the scalar heat equation

Consider the following equation:( ∂

∂t−At

)u(t, x) = f(t, x)

u|t=t0 = 0

(1)

• At is a smooth family of scalar differential operators on the closed manifoldX of order ℓ.


• We impose the following condition on the stabilization of coefficients fort → ∞: A− log(r) is assumed to be C∞ up to r = 0. In particular, Atextends continuously up to t = ∞, and we find a differential operator A∞

of order ℓ there.

Notice that the most natural classical equations fulfill the stabilization conditionimposed on the coefficients. Among these, in particular, there are the autonomousequations, i. e., the coefficients do not depend on time at all, and, moreover,equations that do not depend on time for t≫ t0 sufficiently large.

On the classical notion of parabolicity. Classically, the notion of parabolicity(more precisely: one notion of parabolicity) for equation (1) is strong ellipticity ofthe family of differential operators At, i. e.,

Reσℓψ(At) < 0 on T ∗X \ 0, for all t ∈ [t0,∞]. (2)

Writing the operator ∂t −At from (1) in local coordinates as

∂t −ℓ∑

|α|=0

aα(t, x)Dαx , (3)

the parabolicity condition (2) reads

Re( ∑

|α|=ℓ

aα(t, x)ξα)< 0 (4)

for all x and ξ 6= 0, and all t ∈ [t0,∞].From the local representation (3) the anisotropic structure of the operator

∂t − At is evident. More precisely, it is an operator of order ℓ with the sameanisotropy ℓ, and this is precisely the “gap of orders” of the spatial derivativesand the time derivative. Locally, the anisotropic leading component of the symbolis given as

σℓ;ℓψ

(∂t −

ℓ∑

|α|=0

aα(t, x)Dαx

)(x, t, ξ, ζ) = iζ −

∑

|α|=ℓ

aα(t, x)ξα

according to (3), and we have the anisotropic homogeneity

σℓ;ℓψ

(∂t −

ℓ∑

|α|=0

aα(t, x)Dαx

)(x, t, ξ, ℓζ) = ℓσℓ;ℓψ

(∂t −

ℓ∑

|α|=0

aα(t, x)Dαx

)(x, t, ξ, ζ)

for > 0.Now it is easy to see that the local parabolicity condition (4) is equivalent to

the following:

σℓ;ℓψ

(∂t −

ℓ∑

|α|=0

aα(t, x)Dαx

)(x, t, ξ, ζ) 6= 0 (5)

for all x and t ∈ [t0,∞], and all 0 6= (ξ, ζ) ∈ Rn×H−. Hence we have a conditionon invertibility of the anisotropic principal symbol with the time covariable ζ


extended to the lower half-plane H− = Im(ζ) ≤ 0 ⊆ C (“parabolicity in thesense of Petrovskij”).

The anisotropic homogeneous principal symbol is invariant under changes ofcoordinates, and so is the notion of parabolicity (5), i. e., we require

σℓ;ℓψ(∂t −At)(t, ξx, ζ) 6= 0

for all t ∈ [t0,∞], and (ξx, ζ) ∈(T ∗X×H−

)\ 0.

Change of variables and totally characteristic structure. As we alreadypointed out in the introduction, we shall consider equation (1) not in its originalform, but carry out the change of variables r = e−t to end up with the equation

((−r∂r) −Br

)u(r, x) = f(r, x)

u|r=r0 = 0,

(6)

where Br := A− log(r), which is now considered on the transformed time interval

(0, r0] with r0 := e−t0 . Notice, in particular, that the stabilization condition onthe coefficients now reads that Br is required to be smooth up to r = 0, and thusequation (6) can be regarded as a totally characteristic equation.

Passing to local coordinates as before reveals that the anisotropic leadingcomponent of the symbol of the operator (−r∂r) −Br locally is given as

σℓ;ℓψ

((−r∂r) −

ℓ∑

|α|=0

bα(r, x)Dαx

)(x, r, ξ, ζ) = −irζ −

∑

|α|=ℓ

bα(r, x)ξα.

In particular, we find the typical degenerate structure, and the parabolicity con-dition (5) is equivalent to

σℓ;ℓψ

((−r∂r) −

ℓ∑

|α|=0

bα(r, x)Dαx

)(x, r, ξ, r−1ζ) 6= 0 (7)

for all x, and all 0 6= (ξ, ζ) ∈ Rn×H, and all r ∈ R+. In this condition the upperhalf-plane H is involved instead of the lower half-plane H−, and the degeneracyrequires to consider the above “coupled” expression, which is extended up to theorigin r = 0. The global situation is analogous, i. e., we assume

σℓ;ℓψ((−r∂r) −Br)(r, ξx, r

−1ζ) 6= 0 (8)

for all r ∈ [0, r0], and (ξx, ζ) ∈(T ∗X×H

)\ 0.

Fourier and Mellin representations. We may represent the operator (−r∂r)−Br in the following two ways as a pseudodifferential operator with operator-valuedsymbol:

1) Fourier representation:

(−r∂r) −Br = F−1(−irζ −Br

)F

with the (degenerate) symbol a(r, ζ) := −irζ −Br.


2) Mellin representation:

(−r∂r) −Br = M−1(ζ −Br

)M

with the Mellin symbol h(r, ζ) := ζ −Br.

Notice, in particular, that both symbols a(r, ζ) and h(r, ζ) can be regarded as fami-lies parametrized by the “time variable” r taking values in (anisotropic) parameter-dependent operators on the manifold X (with the anisotropic parameter ζ). Theparabolicity condition (8) in terms of the operator-valued symbols a and h is givenas follows:

1) The parameter-dependent homogeneous principal symbol of a(r, ζ) satisfies

σℓ;ℓψ (a(r, r−1ζ))(ξx, ζ) 6= 0

for all (ξx, ζ) ∈(T ∗X×H

)\ 0, and all r ∈ R+.

2) The parameter-dependent homogeneous principal symbol of h(r, ζ) satisfies

σℓ;ℓψ (h(r, ζ))(ξx, ζ) 6= 0

for all (ξx, ζ) ∈(T ∗X×H0

)\ 0, and all r ∈ R+, where H0 = Re(ζ) ≥ 0 ⊆ C

is the right half-plane.

The Volterra cone calculus makes use of both representations of the operator(−r∂r)−Br. More precisely, the Mellin representation is used close to r = 0, whichcorresponds to t→ ∞ in the original coordinates, while the Fourier representationis used away from r = 0.

Necessary basics of parameter-dependent operators are given in Chapter 3,and the discussion of both the Mellin and Fourier calculus is performed separatelyin Chapter 5. The comments given above for the operator (−r∂r) − Br might beof help to get to a better understanding of their particular structure.

Solvability and regularity. The following result on solvability and regularityof equation (1) is valid under the classical parabolicity condition:

There exists γ0 ∈ R such that for γ > γ0 we have: Given

f ∈ eγt(L2([t0,∞)×X)

),

equation (1) has a unique solution

u ∈ eγt(L2([t0,∞), Hℓ(X))∩H1

0 ([t0,∞), L2(X))).

More precisely, γ0 is given as the spectral bound

γ0 := supRe(λ); λ ∈ spec(A∞)as is visualized in the figure.

Consequently, under the parabolicity assumption, the question whether equa-tion (1) is solvable for all right-hand sides f in some L2-space with fixed exponen-tial weight γ ∈ R depends on whether γ is larger than the spectral bound of the


spec(A∞

)

γ0 γ

Figure 1. The spectrum of A∞

operator A∞. In other words, it is possible if the resolvent (λ −A∞)−1 exists forall Re(λ) ≥ γ.

Let us discuss this in more detail, where we freely interchange between boththe original equation (1) and its transformed representation (6):

In the Mellin representation of the operator (−r∂r) − Br above we met theMellin symbol h(r, ζ) = ζ − Br, and thus the so-called conormal symbol is givenas

σ0M ((−r∂r) −Br)(λ) = h(0, λ) = λ−A∞.

As a consequence of the (classical) parabolicity assumption, the conormal symbolis a parabolic meromorphic Volterra Mellin symbol as discussed in Section 5.2,and consequently is invertible as such. In particular, the resolvent (λ − A∞)−1

exists for Re(λ) sufficiently large, and it is a meromorphic Volterra Mellin symbolof order −ℓ (as a function of λ ∈ C). Notice also that this gives an explanation ofthe figure.

Consequently, equation (1) has a unique solution u for all right-hand sidesf ∈ eγt

(L2([t0,∞)×X)

)for some fixed γ ∈ R if and only if

• the parabolicity condition (8) is fulfilled, i. e., the anisotropic homogeneousprincipal symbol is invertible up to t = ∞,

• the conormal symbol

σ0M ((−r∂r) −Br)(λ) = λ−A∞ : Hs0(X) −→ Hs0−ℓ(X)

is invertible for some s0 ∈ R and all Re(λ) ≥ γ.

Under these conditions, we obtain additionally the above-mentioned maximal reg-ularity assertion, i. e., for the solution u we gain one derivative in time (accordingto the presence of one time derivative in equation (1)), and ℓ derivatives in space(according to the order ℓ of the operators At).


Thus we have seen that the (dominant) symbolic structure of the Volterracone calculus, i. e., the principal symbol and the conormal symbol, both occur inthe discussion of the simple equation (1), and the invertibility of both is requiredto decide about the solvability and regularity in the natural anisotropic Sobolevspaces with an exponential weight, as it is the case for general Volterra coneoperators, too.

We shall not pursue the discussion of the asymptotic behaviour of solutionsof equation (1) here and refer to the main text, in particular, to Chapter 4 forSobolev spaces with asymptotics, and to Chapter 6 for the operator calculus,where equation (1) is a special case as we have seen.

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Institut fur Mathematik, Universitat Potsdam, Postfach 60 15 53, D–14415 Pots-

dam, Germany

E-mail address, T. Krainer: [email protected]

E-mail address, B.–W. Schulze: [email protected]

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